Answer: The graph of is shown. (a) Find and (b) Determine and in terms of (c) Determine

In Exercises 1–6, match the function with one of the graphs [(a), (b), (c), (d), (e), or (f)] using horizontal asymptotes as an aid. \(f(x)=\frac{2 x^{2}}{x^{2}+2}\) Text Transcription: f(x)=2x^2/x^2+2

In Exercises 1–6, match the function with one of the graphs [(a), (b), (c), (d), (e), or (f)] using horizontal asymptotes as an aid. \(f(x)=\frac{2 x}{\sqrt{x^{2}+2}}\) Text Transcription: f(x)=2x/sqrt x^2+2

In Exercises 1–6, match the function with one of the graphs [(a), (b), (c), (d), (e), or (f)] using horizontal asymptotes as an aid. \(f(x)=\frac{x}{x^{2}+2}\) Text Transcription: f(x)=x/x^2+2

In Exercises 1–6, match the function with one of the graphs [(a), (b), (c), (d), (e), or (f)] using horizontal asymptotes as an aid. \(f(x)=2+\frac{x^{2}}{x^{4}+1}\) Text Transcription: f(x)=2+x^2/x^4+1

In Exercises 1–6, match the function with one of the graphs [(a), (b), (c), (d), (e), or (f)] using horizontal asymptotes as an aid. \(f(x)=\frac{4 \sin x}{x^{2}+1}\) Text Transcription: f(x)=4 sin x/x^2+1

In Exercises 1–6, match the function with one of the graphs [(a), (b), (c), (d), (e), or (f)] using horizontal asymptotes as an aid. \(f(x)=\frac{2 x^{2}-3 x+5}{x^{2}+1}\) Text Transcription: f(x)=2x^2-3x+5/x^2+1

Numerical and Graphical Analysis In Exercises 7–12, use a graphing utility to complete the table and estimate the limit as x approaches infinity. Then use a graphing utility to graph the function and estimate the limit graphically. \(f(x)=\frac{4 x+3}{2 x-1}\) Text Transcription: f(x)=4x+3/2x-1

Numerical and Graphical Analysis In Exercises 7–12, use a graphing utility to complete the table and estimate the limit as x approaches infinity. Then use a graphing utility to graph the function and estimate the limit graphically. \(f(x)=\frac{2 x^{2}}{x+1}\) Text Transcription: f(x)=2x^2/x+1

Numerical and Graphical Analysis In Exercises 7–12, use a graphing utility to complete the table and estimate the limit as x approaches infinity. Then use a graphing utility to graph the function and estimate the limit graphically. \(f(x)=\frac{-6 x}{\sqrt{4 x^{2}+5}}\) Text Transcription: f(x)=-6x/sqrt 4x^2+5

Numerical and Graphical Analysis In Exercises 7–12, use a graphing utility to complete the table and estimate the limit as x approaches infinity. Then use a graphing utility to graph the function and estimate the limit graphically. \(f(x)=\frac{20 x}{\sqrt{9 x^{2}-1}}\) Text Transcription: f(x)=20x/sqrt 9x^2-1

Numerical and Graphical Analysis In Exercises 7–12, use a graphing utility to complete the table and estimate the limit as x approaches infinity. Then use a graphing utility to graph the function and estimate the limit graphically. \(f(x)=5-\frac{1}{x^{2}+1}\) Text Transcription: f(x)=5-1/x^2+1

Numerical and Graphical Analysis In Exercises 7–12, use a graphing utility to complete the table and estimate the limit as x approaches infinity. Then use a graphing utility to graph the function and estimate the limit graphically. \(f(x)=4+\frac{3}{x^{2}+2}\) Text Transcription: f(x)=4+3/x^2+2

In Exercises 13 and 14 , find \(\lim _{x \rightarrow \infty} h(x)\), if possible. \(f(x)=5 x^{3}-3 x^{2}+10 x\) (a) \(h(x)=\frac{f(x)}{x^{2}}\) (b) \(h(x)=\frac{f(x)}{x^{3}}\) (c) \(h(x)=\frac{f(x)}{x^{4}}\) Text Transcription: lim _x rightarrow infty h(x) h(x)=f(x)/x^2 h(x)=f(x)/x^3 h(x)=f(x)/x^4

In Exercises 13 and 14 , find \(\lim _{x \rightarrow \infty} h(x)\), if possible. \(f(x)=-4 x^{2}+2 x-5\) (a) \(h(x)=\frac{f(x)}{x}\) (b) \(h(x)=\frac{f(x)}{x^{2}}\) (c) \(h(x)=\frac{f(x)}{x^{3}}\) Text Transcription: lim _x rightarrow infty h(x) h(x)=f(x)/x h(x)=f(x)/x^2 h(x)=f(x)/x^3

In Exercises 15–18, find each limit, if possible. (a) \(\lim _{x \rightarrow \infty} \frac{x^{2}+2}{x^{3}-1}\) (b) \(\lim _{x \rightarrow \infty} \frac{x^{2}+2}{x^{2}-1}\) (c) \(\lim _{x \rightarrow \infty} \frac{x^{2}+2}{x-1}\) Text Transcription: lim _x rightarrow infty x^2+2/x^3-1 lim _x rightarrow infty x^2+2/x^2-1 lim _x rightarrow infty x^2+2/x-1

In Exercises 15-18, find each limit, if possible. (a) \(\lim _{x \rightarrow \infty} \frac{3-2 x}{3 x^{3}-1}\) (b) \(\lim _{x \rightarrow \infty} \frac{3-2 x}{3 x-1}\) (c) \(\lim _{x \rightarrow \infty} \frac{3-2 x^{2}}{3 x-1}\) Text Transcription: lim_x rightarrow infinity 3-2x/3x^3-1 lim_x rightarrow infinity 3-2x/3x-1 lim_x rightarrow infinity 3-2x^2/3x-1

In Exercises 15-18, find each limit, if possible. (a) \(\lim _{x \rightarrow \infty} \frac{5-2 x^{3 / 2}}{3 x^{2}-4}\) (b) \(\lim _{x \rightarrow \infty} \frac{5-2 x^{3 / 2}}{3 x^{3 / 2}-4}\) (c) \(\lim _{x \rightarrow \infty} \frac{5-2 x^{3 / 2}}{3 x-4}\) Text Transcription: lim_x rightarrow infinity 5-2x^3/2/3x^2-4 lim_x rightarrow infinity 5-2x^3/2/3x^3/2-4 lim_x rightarrow infinity 5-2x^3/2/3x-4

In Exercises 15-18, find each limit, if possible. (a) \(\lim _{x \rightarrow \infty} \frac{5 x^{3 / 2}}{4 x^{2}+1}\) (b) \(\lim _{x \rightarrow \infty} \frac{5 x^{3 / 2}}{4 x^{3 / 2}+1}\) (c) \(\lim _{x \rightarrow \infty} \frac{5 x^{3 / 2}}{4 \sqrt{x}+1}\) Text Transcription: lim_x rightarrow infinity 5x^3/2/4x^2+1 lim_x rightarrow infinity 5x^3/2/4x^3/2+1 lim_x rightarrow infinity 5x^3/2/4 sqrt x + 1

In Exercises 19-38, find the limit. \(\lim _{x \rightarrow \infty}\left(4+\frac{3}{x}\right)\) Text Transcription: lim_x rightarrow infinity (4+3/x)

In Exercises 19-38, find the limit. \(\lim _{x \rightarrow-\infty}\left(\frac{5}{x}-\frac{x}{3}\right)\) Text Transcription: lim_x rightarrow infinity (5/x-x/3)

In Exercises 19-38, find the limit. \(\lim _{x \rightarrow \infty} \frac{2 x-1}{3 x+2}\) Text Transcription: lim_x rightarrow infinity 2x-1/3x+2

In Exercises 19-38, find the limit. \(\lim _{x \rightarrow \infty} \frac{x^{2}+3}{2 x^{2}-1}\) Text Transcription: lim_x rightarrow infinity x^2+3/2x^2-1

In Exercises 19-38, find the limit. \(\lim _{x \rightarrow \infty} \frac{x}{x^{2}-1}\) Text Transcription: lim_x rightarrow infinity x/x^2-1

In Exercises 19-38, find the limit. \(\lim _{x \rightarrow \infty} \frac{5 x^{3}+1}{10 x^{3}-3 x^{2}+7}\) Text Transcription: lim_x rightarrow infinity 5x^3+1/10x^3-3x^2+7

In Exercises 19-38, find the limit. \(\lim _{x \rightarrow-\infty} \frac{5 x^{2}}{x+3}\) Text Transcription: lim_x rightarrow -infinity 5x^2/x+3

In Exercises 19-38, find the limit. \(\lim _{x \rightarrow-\infty}\left(\frac{1}{2} x-\frac{4}{x^{2}}\right)\) Text Transcription: lim_x rightarrow -infinity (1/2x-4/x^2)

In Exercises 19-38, find the limit. \(\lim _{x \rightarrow-\infty} \frac{x}{\sqrt{x^{2}-x}}\) Text Transcription: lim_x rightarrow -infinity x/sqrt x^2-x

In Exercises 19-38, find the limit. \(\lim _{x \rightarrow-\infty} \frac{x}{\sqrt{x^{2}+1}}\) Text Transcription: lim_x rightarrow -infinity x/sqrt x^2+1

In Exercises 19-38, find the limit. \(\lim _{x \rightarrow-\infty} \frac{2 x+1}{\sqrt{x^{2}-x}}\) Text Transcription: lim_x rightarrow -infinity 2x+1/sqrt x^2-x

In Exercises 19-38, find the limit. \(\lim _{x \rightarrow-\infty} \frac{-3 x+1}{\sqrt{x^{2}+x}}\) Text Transcription: lim_x rightarrow -infinity -3x+1/sqrt x^2+x

In Exercises 19-38, find the limit. \(\lim _{x \rightarrow \infty} \frac{\sqrt{x^{2}-1}}{2 x-1}\) Text Transcription: lim_x rightarrow infinity sqrt x^2-1/2x-1

In Exercises 19-38, find the limit. \(\lim _{x \rightarrow-\infty} \frac{\sqrt{x^{4}-1}}{x^{3}-1}\) Text Transcription: lim_x rightarrow -infinity sqrt x^4-1/x^3-1

In Exercises 19-38, find the limit. \(\lim _{x \rightarrow \infty} \frac{x+1}{\left(x^{2}+1\right)^{1 / 3}}\) Text Transcription: lim_x rightarrow infinity x+1/(x^2+1)^1/3

In Exercises 19-38, find the limit. \(\lim _{x \rightarrow-\infty} \frac{2 x}{\left(x^{6}-1\right)^{1 / 3}}\) Text Transcription: lim_x rightarrow -infinity 2x/(x^6-1)^1/3

In Exercises 19-38, find the limit. \(\lim _{x \rightarrow \infty} \frac{1}{2 x+\sin x}\) Text Transcription: lim_x rightarrow infinity 1/2x+sin x

In Exercises 19-38, find the limit. \(\lim _{x \rightarrow \infty} \cos \frac{1}{x}\) Text Transcription: lim_x rightarrow infinity cos 1/x

In Exercises 19-38, find the limit. \(\lim _{x \rightarrow \infty} \frac{\sin 2 x}{x}\) Text Transcription: lim_x rightarrow infinity sin 2x/x

In Exercises 19-38, find the limit. \(\lim _{x \rightarrow \infty} \frac{x-\cos x}{x}\) Text Transcription: lim_x rightarrow infinity x-cos-x/x

In Exercises 39-42, use a graphing utility to graph the function and identify any horizontal asymptotes. \(f(x)=\frac{|x|}{x+1}\) Text Transcription: f(x)=|x|/x+1

In Exercises 39-42, use a graphing utility to graph the function and identify any horizontal asymptotes. \(f(x)=\frac{|3 x+2|}{x-2}\) Text Transcription: f(x)=|3x+2|/x-2

In Exercises 39-42, use a graphing utility to graph the function and identify any horizontal asymptotes. \(f(x)=\frac{3 x}{\sqrt{x^{2}+2}}\) Text Transcription: f(x)=3x/sqrt x^2+2

In Exercises 39-42, use a graphing utility to graph the function and identify any horizontal asymptotes. \(f(x)=\frac{\sqrt{9 x^{2}-2}}{2 x+1}\) Text Transcription: f(x)=sqrt 9x^2-2/2x+1

In Exercises 43 and 44, find the limit. (Hint: Let x = 1/t and find the limit as \(t \rightarrow 0^{+}\).) \(\lim _{x \rightarrow \infty} x \sin \frac{1}{x}\) Text Transcription: t rightarrow 0^+ lim_x rightarrow infinity x sin 1/x

In Exercises 43 and 44, find the limit. (Hint: Let x = 1/t and find the limit as \(t \rightarrow 0^{+}\).) \(\lim _{x \rightarrow \infty} x \tan \frac{1}{x}\) Text Transcription: t rightarrow 0^+ lim_x rightarrow infinity x tan 1/x

In Exercises 45-48, find the limit. (Hint: Treat the expression as a fraction whose denominator is 1, and rationalize the numerator.) Use a graphing utility to verify your result. \(\lim _{x \rightarrow-\infty}\left(x+\sqrt{x^{2}+3}\right)\) Text Transcription: lim_x rightarrow -infinity (x+sqrt x^2+3)

In Exercises 45-48, find the limit. (Hint: Treat the expression as a fraction whose denominator is 1, and rationalize the numerator.) Use a graphing utility to verify your result. \(\lim _{x \rightarrow \infty}\left(x-\sqrt{x^{2}+x}\right)\) Text Transcription: lim_x rightarrow infinity (x-sqrt x^2+x)

In Exercises 45-48, find the limit. (Hint: Treat the expression as a fraction whose denominator is 1, and rationalize the numerator.) Use a graphing utility to verify your result. \(\lim _{x \rightarrow-\infty}\left(3 x+\sqrt{9 x^{2}-x}\right)\) Text Transcription: lim_x rightarrow -infinity (3x+sqrt 9x^2-x)

In Exercises 45-48, find the limit. (Hint: Treat the expression as a fraction whose denominator is 1, and rationalize the numerator.) Use a graphing utility to verify your result. \(\lim _{x \rightarrow \infty}\left(4 x-\sqrt{16 x^{2}-x}\right)\) Text Transcription: lim_x rightarrow infinity (4x-sqrt 16x^2-x)

Numerical, Graphical, and Analytic Analysis In Exercises 49–52, use a graphing utility to complete the table and estimate the limit as x approaches infinity. Then use a graphing utility to graph the function and estimate the limit. Finally, find the limit analytically and compare your results with the estimates \(f(x)=x-\sqrt{x(x-1)}\) Text Transcription: f(x)=x-sqrt x(x-1)

Numerical, Graphical, and Analytic Analysis In Exercises 49–52, use a graphing utility to complete the table and estimate the limit as x approaches infinity. Then use a graphing utility to graph the function and estimate the limit. Finally, find the limit analytically and compare your results with the estimates \(f(x)=x^{2}-x \sqrt{x(x-1)}\) Text Transcription: f(x)=x^2-x sqrt x(x-1)

Numerical, Graphical, and Analytic Analysis In Exercises 49–52, use a graphing utility to complete the table and estimate the limit as x approaches infinity. Then use a graphing utility to graph the function and estimate the limit. Finally, find the limit analytically and compare your results with the estimates \(f(x)=x \sin \frac{1}{2 x}\) Text Transcription: f(x)=x sin 1/2x

Numerical, Graphical, and Analytic Analysis In Exercises 49–52, use a graphing utility to complete the table and estimate the limit as x approaches infinity. Then use a graphing utility to graph the function and estimate the limit. Finally, find the limit analytically and compare your results with the estimates \(f(x)=\frac{x+1}{x \sqrt{x}}\) Text Transcription: f(x)=x+1/x sqrt x

In Exercises 53 and 54, describe in your own words what the statement means. \(\lim _{x \rightarrow \infty} f(x)=4\) Text Transcription: lim_x rightarrow infinity f(x)=4

In Exercises 53 and 54, describe in your own words what the statement means. \(\lim _{x \rightarrow-\infty} f(x)=2\) Text Transcription: lim_x rightarrow -infinity f(x)=2

Sketch a graph of a differentiable function f that satisfies the following conditions and has x = 2 as its only eritical number. \(\begin{array}{l} f^{\prime}(x)<0 \text { for } x<2 \quad f^{\prime}(x)>0 \text { for } x>2 \\ \lim _{x \rightarrow-\infty} f(x)=\lim _{x \rightarrow \infty} f(x)=6 \end{array} \) Text Transcription: f’(x)<0 for x < 2 f’(x)>0 for x > 2 lim_x rightarrow -infinity f(x)=lim_x rightarrow f(x)=6

Is it possible to sketch a graph of a function that satisfies the conditions of Exercise 55 and has no points of inflection? Explain.

If f is a continuous function such that lim \(\\lim _{x \rightarrow \infty} f(x)=5), find, if possible, \(\lim _{x \rightarrow-\infty} f(x)\) for each specified condition. (a) The graph of f is symmetric with respect to the y-axis. (b) The graph of f is symmetric with respect to the origin. Text Transcription: lim_x rightarrow infinityf(x)=5 lim_x rightarrow =infinity f(x)

The graph of a function f is shown below. To print an enlarged copy of the graph, go to the website www.mathgraphs.com. (a) Sketch f'. (b) Use the graphs to estimate \(\lim _{x \rightarrow \infty} f(x)\) and \(\lim _{x \rightarrow \infty} f^{\prime}(x)\). (c) Explain the answers you gave in part (b). Text Transcription: lim_x righarrow infinity f(x) lim_x righarrow infinity f’(x)

In Exercises 59-76, sketch the graph of the equation using extrema, intercepts, symmetry, and asymptotes. Then use a graphing utility to verify your result. \(y=\frac{x}{1-x}\) Text Transcription: y=x/1-x

In Exercises 59-76, sketch the graph of the equation using extrema, intercepts, symmetry, and asymptotes. Then use a graphing utility to verify your result. \(y=\frac{x-4}{x-3}\) Text Transcription: y=x-4/x-3

In Exercises 59-76, sketch the graph of the equation using extrema, intercepts, symmetry, and asymptotes. Then use a graphing utility to verify your result. \(y=\frac{x+1}{x^{2}-4}\) Text Transcription: y=x+1/x^2-4

In Exercises 59-76, sketch the graph of the equation using extrema, intercepts, symmetry, and asymptotes. Then use a graphing utility to verify your result. \(y=\frac{2 x}{9-x^{2}}\) Text Transcription: y=2x/9-x^2

In Exercises 59-76, sketch the graph of the equation using extrema, intercepts, symmetry, and asymptotes. Then use a graphing utility to verify your result. \(y=\frac{x^{2}}{x^{2}+16}\) Text Transcription: y=x^2/x^2+16

In Exercises 59-76, sketch the graph of the equation using extrema, intercepts, symmetry, and asymptotes. Then use a graphing utility to verify your result. \(y=\frac{x^{2}}{x^{2}-16}\) Text Transcription: y=x^2/x^2-16

In Exercises 59-76, sketch the graph of the equation using extrema, intercepts, symmetry, and asymptotes. Then use a graphing utility to verify your result. \(y=\frac{2 x^{2}}{x^{2}-4}\) Text Transcription: y=2x^2/x^2-4

In Exercises 59-76, sketch the graph of the equation using extrema, intercepts, symmetry, and asymptotes. Then use a graphing utility to verify your result. \(y=\frac{2 x^{2}}{x^{2}+4}\) Text Transcription: y=2x^2/x^2+4

In Exercises 59-76, sketch the graph of the equation using extrema, intercepts, symmetry, and asymptotes. Then use a graphing utility to verify your result. \(x y^{2}=9\) Text Transcription: xy^2=9

In Exercises 59-76, sketch the graph of the equation using extrema, intercepts, symmetry, and asymptotes. Then use a graphing utility to verify your result. \(x^{2} y=9\) Text Transcription: x^2y=9

In Exercises 59-76, sketch the graph of the equation using extrema, intercepts, symmetry, and asymptotes. Then use a graphing utility to verify your result. \(y=\frac{3 x}{1-x}\) Text Transcription: y=3x/1-x

In Exercises 59-76, sketch the graph of the equation using extrema, intercepts, symmetry, and asymptotes. Then use a graphing utility to verify your result. \(y=\frac{3 x}{1-x^{2}}\) Text Transcription: y=3x/1-x^2

In Exercises 59-76, sketch the graph of the equation using extrema, intercepts, symmetry, and asymptotes. Then use a graphing utility to verify your result. \(y=2-\frac{3}{x^{2}}\) Text Transcription: y=2-3/x^2

In Exercises 59-76, sketch the graph of the equation using extrema, intercepts, symmetry, and asymptotes. Then use a graphing utility to verify your result. \(y=1+\frac{1}{x}\) Text Transcription: y=1+1/x

In Exercises 59-76, sketch the graph of the equation using extrema, intercepts, symmetry, and asymptotes. Then use a graphing utility to verify your result. \(y=3+\frac{2}{x}\) Text Transcription: y=3+2/x

In Exercises 59-76, sketch the graph of the equation using extrema, intercepts, symmetry, and asymptotes. Then use a graphing utility to verify your result. \(y=4\left(1-\frac{1}{x^{2}}\right)\) Text Transcription: y=4(1-1/x^2)

In Exercises 59-76, sketch the graph of the equation using extrema, intercepts, symmetry, and asymptotes. Then use a graphing utility to verify your result. \(y=\frac{x^{3}}{\sqrt{x^{2}-4}}\) Text Transcription: y=x^3/sqrt x^2-4

In Exercises 59-76, sketch the graph of the equation using extrema, intercepts, symmetry, and asymptotes. Then use a graphing utility to verify your result. \(y=\frac{x}{\sqrt{x^{2}-4}}\) Text Transcription: y=x/sqrt x^2-4

In Exercises 77-84, use a computer algebra system to analyze the graph of the function. Label any extrema and/or asymptotes that exist. \(f(x)=9-\frac{5}{x^{2}}\) Text Transcription: f(x)=9-5/x^2

In Exercises 77-84, use a computer algebra system to analyze the graph of the function. Label any extrema and/or asymptotes that exist. \(f(x)=\frac{1}{x^{2}-x-2}\) Text Transcription: f(x)=1/x^2-x-2

In Exercises 77-84, use a computer algebra system to analyze the graph of the function. Label any extrema and/or asymptotes that exist. \(f(x)=\frac{x-2}{x^{2}-4 x+3}\) Text Transcription: f(x)=x-2/x^2-4x+3

In Exercises 77-84, use a computer algebra system to analyze the graph of the function. Label any extrema and/or asymptotes that exist. \(f(x)=\frac{x+1}{x^{2}+x+1}\) Text Transcription: f(x)=x+1/x^2+x+1

In Exercises 77-84, use a computer algebra system to analyze the graph of the function. Label any extrema and/or asymptotes that exist. \(f(x)=\frac{3 x}{\sqrt{4 x^{2}+1}}\) Text Transcription: f(x)=3x/sqrt 4x^2+1

In Exercises 77-84, use a computer algebra system to analyze the graph of the function. Label any extrema and/or asymptotes that exist. \(g(x)=\frac{2 x}{\sqrt{3 x^{2}+1}}\) Text Transcription: g(x)=2x/sqrt 3x^2+1

In Exercises 77-84, use a computer algebra system to analyze the graph of the function. Label any extrema and/or asymptotes that exist. \(g(x)=\sin \left(\frac{x}{x-2}\right), \quad x>3\) Text Transcription: g(x)=sin(x/x-2), x>3

In Exercises 77-84, use a computer algebra system to analyze the graph of the function. Label any extrema and/or asymptotes that exist. \(f(x)=\frac{2 \sin 2 x}{x}\) Text Transcription: f(x)=2 sin 2x/x

In Exercises 85 and 86, (a) use a graphing utility to graph f and g in the same viewing window, (b) verify algebraically that f and g represent the same function, and (c) zoom out sufficiently far so that the graph appears as a line. What equation does this line appear to have? (Note that the points at which the function is not continuous are not readily seen when you zoom out.) \(\begin{array}{l} f(x)=\frac{x^{3}-3 x^{2}+2}{x(x-3)} \\ g(x)=x+\frac{2}{x(x-3)} \end{array}\) Text Transcription: f(x)=x^3-3x^2+2/x(x-3) g(x)=x+2/x(x-3)

In Exercises 85 and 86, (a) use a graphing utility to graph f and g in the same viewing window, (b) verify algebraically that f and g represent the same function, and (c) zoom out sufficiently far so that the graph appears as a line. What equation does this line appear to have? (Note that the points at which the function is not continuous are not readily seen when you zoom out.) \(\begin{array}{l}f(x)=-\frac{x^{3}-2 x^{2}+2}{2 x^{2}} \\g(x)=-\frac{1}{2} x+1-\frac{1}{x^{2}}\end{array}\) Text Transcription: f(x)=-x^3-2x^2+2/2x^2 g(x)=-1/2x+1-1/x^2

Engine Efficiency The efficiency of an internal combustion engine is Efficiency (%) = \(100\left[1-\frac{1}{\left(v_{1} / v_{2}\right)^{c}}\right]\) where \(v_{1} / v_{2}\) is the ratio of the uncompressed gas to he compressed gas and c is a positive constant dependent on the engine design. Find the limit of the efficiency as the compression ratio approaches infinity. Text Transcription: 100[1-1/(v_1/v_2)^c] v_1/v_2

Average Cost A business has a cost of C = 0.5x + 500 for producing x units. The average cost per unit is \(\bar{C}=\frac{C}{x}\). Find the limit of \(\bar{C}\) as x approaches infinity. Text Transcription: bar C = C/x bar C

Physics Newton's First Law of Motion and Einstein's Special Theory of Relativity differ concerning a particle's behavior as its velocity approaches the speed of light c. In the graph, functions N and E represent the velocity v, with respect to time t, of a particle accelerated by a constant force as predicted by Newton and Einstein. Write limit statements that describe these two theories

Temperature The graph shows the temperature T, in degrees Fahrenheit, of molten glass t seconds after it is removed from a kiln. (a) Find \(\lim _{t \rightarrow 0^{+}}\) T. What does this limit represent? (b) Find \(\lim _{t \rightarrow \infty}\) T. What does this limit represent? (c) Will the temperature of the glass ever actually reach room temperature? Why? Text Transcription: lim_t rightarrow 0^+ lim_t rightarrow infinity

Modeling Data The table shows the world record times for the mile run, where t represents the year, with t = 0 corresponding to 1900, and y is the time in minutes and seconds. A model for the data is \(y=\frac{3.351 t^{2}+42.461 t-543.730}{t^{2}}\) where the seconds have been changed to decimal parts of a minute. (a) Use a graphing utility to plot the data and graph the model. (b) Does there appear to be a limiting time for running 1 mile? Explain. Text Transcription: y=3.351 t^2+42.461 t-543.730/t^2

Modeling Data The average typing speeds S (in words per minute) of a typing student after t weeks of lessons are shown in the table A model for the data is \(S=\frac{100 t^{2}}{65+t^{2}}, \quad t>0\). (a) Use a graphing utility to plot the data and graph the model. (b) Does there appear to be a limiting typing speed? Explain. Text Transcription: S=100t^2/65+t^2, t>0

Modeling Data A heat probe is attached to the heat exchanger of a heating system. The temperature T (in degrees Celsius) is recorded t seconds after the furnace is started. The results for the first 2 minutes are recorded in the table. (a) Use the regression capabilities of a graphing utility to find a model of the form \(T_{1}=a t^{2}+b t+c\) for the data. (b) Use a graphing utility to graph \(T_{1}\). (c) A rational model for the data is \(T_{2}=\frac{1451+86 t}{58+t}\). Use a graphing utility to graph \(T_{2}\). (d) Find \(T_{1}(0)\) and \(T_{2}(0)\). (e) Find \(\lim _{t \rightarrow \infty} T_{2}\) (f) Interpret the result in part (e) in the context of the problem. Is it possible to do this type of analysis using \(T_{1}\)? Explain. Text Transcription: T_1=at^2+bt+c T_1 T_2=1451+86t/58+t T_2 T_1(0) T_2(0) lim_t rightarrow infinity T_2

Modeling Data A container holds 5 liters of a 25% brine solution. The table shows the concentrations C of the mixture after adding x liters of a 75% brine solution to the container. (a) Use the regression features of a graphing utility to find a model of the form \(C_{1}=a x^{2}+b x+c\) for the data. (b) Use a graphing utility to graph \(C_{1}\). (c) A rational model for these data is \(C_{2}=\frac{5+3 x}{20+4 x}\).Use a graphing utility to graph \(C_{2}\). (d) Find \(\lim _{x \rightarrow \infty} C_{1}\) and \(\lim _{x \rightarrow \infty} C_{2}\). Which model do you think best represents the concentration of the mixture? Explain. Text Transcription: C_1=a x^2+b x+c C_1 C_2=5+3x/20+4x C_2 lim_x rightarrow infinity C_1 lim_x rightarrow infinity C_2

A line with slope m passes through the point (0, 4). (a) Write the distance d between the line and the point (3, 1) as a function of m. (b) Use a graphing utility to graph the equation in part (a). (c) Find \(\lim _{m \rightarrow \infty} d(m)\) and \(\lim _{m \rightarrow -\infty} d(m)\). Interpret the results geometrically. Text Transcription: lim_m rightarrow infinity d(m) lim_m rightarrow -infinity d(m)

A line with slope m passes through the point (0, -2). (a) Write the distance d between the line and the point (4, 2) as a function of m. (b) Use a graphing utility to graph the equation in part (a). (c) Find \(\lim _{m \rightarrow \infty} d(m)\) and \(\lim _{m \rightarrow -\infty} d(m)\). Interpret the results geometrically. Text Transcription: lim_m rightarrow infinity d(m) lim_m rightarrow -infinity d(m)

The graph of \(f(x)=\frac{2 x^{2}}{x^{2}+2}\) is shown. (a) Find \(L=\lim _{x \rightarrow \infty} f(x)\). (b) Determine \(x_{1}\) and \(x_{2}\) in terms of \(\varepsilon\). (c) Determine M, where M > 0, such that |f(x) - L| < \(\varepsilon\) for x > M. (d) Determine N, where N < 0, such that |f(x) - L| < \(\varepsilon\) for x < N. Text Transcription: f(x)=2x^2/x^2+2 L=lim_x rightarrow infinity f(x) x_1 x_2 epsilon

The graph of \(f(x)=\frac{6 x}{\sqrt{x^{2}+2}}\) is shown. (a) Find \(L=\lim _{x \rightarrow \infty} f(x)\) and \(K=\lim _{x \rightarrow -\infty} f(x)\). (b) Determine \(x_{1}\) and \(x_{2}\) in terms of \(\varepsilon\).. (c) Determine M, where M > 0, such that |f(x) - L| < \(\varepsilon\) for x > M. (d) Determine N, where N < 0, such that |f(x) - K| < \(\varepsilon\) for x < N. Text Transcription: f(x)=6x/sqrt x^2+2 L=lim_x rightarrow infinity f(x) K=lim_x rightarrow -infinity f(x) x_1 x_2 epsilon

Consider \(\lim _{x \rightarrow \infty} \frac{3 x}{\sqrt{x^{2}+3}}\). Use the definition of limits at infinity to find values of M that correspond to (a) \(\varepsilon=0.5\) and (b) \(\varepsilon=0.1\). Text Transcription: lim_x rightarrow infinity 3x/sqrt x^2+3 epsilon=0.5 epsilon=0.1

Consider \(\lim _{x \rightarrow -\infty} \frac{3 x}{\sqrt{x^{2}+3}}\). Use the definition of limits at infinity to find values of M that correspond to (a) \(\varepsilon=0.5\) and (b) \(\varepsilon=0.1\). Text Transcription: lim_x rightarrow -infinity 3x/sqrt x^2+3 epsilon=0.5 epsilon=0.1

In Exercises 101-104, use the definition of limits at infinity to prove the limit. \(\lim _{x \rightarrow \infty} \frac{1}{x^{2}}=0\) Text Transcription: lim_x rightarrow infinity 1/x^2=0

In Exercises 101-104, use the definition of limits at infinity to prove the limit. \(\lim _{x \rightarrow \infty} \frac{2}{\sqrt{x}}=0\) Text Transcription: lim_x rightarrow infinity 2/sqrt x=0

In Exercises 101-104, use the definition of limits at infinity to prove the limit. \(\lim _{x \rightarrow-\infty} \frac{1}{x^{3}}=0\) Text Transcription: lim_x rightarrow -infinity 1/x^3=0

In Exercises 101-104, use the definition of limits at infinity to prove the limit. \(\lim _{x \rightarrow-\infty} \frac{1}{x-2}=0\) Text Transcription: lim_x rightarrow -infinity 1/x-2=0

Prove that if \(p(x)=a_{n} x^{n}+\cdots+a_{1} x+a_{0}\) and \(q(x)=b_{m} x^{m}+\cdots+b_{1} x+b_{0}\left(a_{n} \neq 0, b_{m} \neq 0\right)\), then \(\lim _{x \rightarrow \infty} \frac{p(x)}{q(x)}=\left\{\begin{array}{ll} 0, & n<m \\ \frac{a_{n}}{b_{m}}, & n=m \\ \pm \infty, & n>m \end{array}\right.\). Text Transcription: p(x)=a_nx^n+...+a_1x+a_0 q(x)=b_mx^m+...+b_1x+b_0 (a_n neq 0, b_m neq 0) lim_x rightarrow infinity p(x)/q(x)={0, n<m a_n/b_m, n=m +- infinity, n>m

Use the definition of infinite limits at infinity to prove that \(\lim _{x \rightarrow \infty} x^{3}=\infty\). Text Transcription: lim_x rightarrow infinity x^3=infinity

True or false? In Exercises 107 and 108, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If f’(x) > 0 for all real numbers x, then f increases without bound.

True or false? In Exercises 107 and 108, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If f"(x) < 0 for all real numbers x, then f decreases without bound.