?Let\(g(x)=\left\{\begin{array}{ll} 5 x-2 & \text { if } x<1 \\ a & \text { if } x=1 \\

Determine whether the following statements are true and give an explanation or counterexample. a. The rational function \(\frac{x-1}{x^2-1}\) has vertical asymptotes at x = - 1 and x = 1. b. Numerical or graphical methods always produce good estimates of \(\lim _{x \rightarrow a} f(x)\). c. The value of \(\lim _{x \rightarrow a} f(x)\), if it exists, is found by calculating f(a), d. If \(\lim _{x \rightarrow a} f(x)=\infty\) or \(\lim _{x \rightarrow a} f(x)=-\infty\), then \(\lim _{x \rightarrow a} f(x)\) does not exists. e. If \(\lim _{x \rightarrow a} f(x)\) does not exists, then either \(\lim _{x \rightarrow a} f(x)=\infty\) or \(\lim _{x \rightarrow a} f(x)=-\infty\). f. If a function is continuous on the intervals (a, b) and [b, c), where a < b < c, then the function is also continuous on (a, c). g. If \(\lim _{x \rightarrow a} f(x)\) can be calculated by direct substitution, then f is continuous at x = a.

Use the graph of f in the figure to find the following values, if possible. a. f(-1) b. \(\lim _{x \rightarrow -1^-} f(x)\) c. \(\lim _{x \rightarrow -1^+} f(x)\) d. \(\lim _{x \rightarrow -1} f(x)\) e. f(1) f. \(\lim _{x \rightarrow 1} f(x)\) g. \(\lim _{x \rightarrow 2} f(x)\) h. \(\lim _{x \rightarrow 3^-} f(x)\) i. \(\lim _{x \rightarrow 3^+} f(x)\) j. \(\lim _{x \rightarrow 3} f(x)\)

Use the graph of f in the figure to determine the values of x in the interval (-3, 5) at which f fails to be continuous. Justify your answers using the continuity checklist.

Computing a limit graphically and analytically a. Graph \(y=\frac{2 \theta}{sin\ \theta}\). Comment on any inaccuracies in the graph and then sketch an accurate graph of the function. b. Estimate \(\lim _{\theta \rightarrow 0} \frac{sin 2 \theta}{sin \theta}\) using the graph in part (a). c. Verify your answer to part (b) by finding the value of \(\lim _{\theta \rightarrow 0} \frac{sin 2 \theta}{sin \theta}\) analytically using the trigonometric identity \(\sin2 \theta=2\sin \theta\cos \theta\).

Computing a limit numerically and analytically a. Estimate \(\lim _{x \rightarrow \pi/4} \frac{cos 2x}{cos x - sin x}\) by making a table of values of \(\frac{cos 2x}{cos x-sin x}\) for values of x approaching \(\pi/4\). Round your estimate to four digits. b. Use analytic methods to find the value of \(\lim _{x \rightarrow \pi/4} \frac{cos 2x}{cos x - sin x}\).

Suppose a long-distance phone call costs $0.75 for the first min (or any part of the first min), plus $0.10 for each additional min (or any part of a min). a. Graph the function c = f(t) that gives the cost for talking on the phone for t minutes for \(0\ \leq\ t\ \leq\ 5\). b. Evaluate \(\lim _{t \rightarrow 2.9} f(t)\). c. Evaluate \(\lim _{t \rightarrow 3^-} f(t)\) and \(\lim _{t \rightarrow 3^+} f(t)\). d. Interpret the meaning of the limits in part (c). e. For what values of t is f continuous? Explain.

Sketch the graph of a function f with all the following properties. \(\lim _{x \rightarrow 2^-} f(x)=\infty\) \(\lim _{x \rightarrow 2^+} f(x)=-\infty\) \(\lim _{x \rightarrow 0} f(x)=\infty\) \(\lim _{x \rightarrow 3^-} f(x)=2\) \(\lim _{x \rightarrow 3^+} f(x)=4\) f(3) = 1

Calculate the following limits analytically. \(\lim _{x \rightarrow 1000} 18\pi^2\)

Calculate the following limits analytically. \(\lim _{x \rightarrow 1} \sqrt{5x+6}\)

Calculate the following limits analytically. \(\lim _{h \rightarrow 0} \frac{\sqrt{5x+5h}-\sqrt{5x}}{h}\), where x is constant

Calculate the following limits analytically. \(\lim _{x \rightarrow 1} \frac{x^3-7x^2+12x}{4-x}\)

Calculate the following limits analytically. \(\lim _{x \rightarrow 4} \frac{x^3 -7x^2+12x}{4-x}\)

Calculate the following limits analytically. \(\lim _{x \rightarrow 1} \frac{1-x^2}{x^2-8x+7}\)

Calculate the following limits analytically. \(\lim _{x \rightarrow 3} \frac{\sqrt{3x+16}-5}{x-3}\)

Calculate the following limits analytically. \(\lim _{x \rightarrow 3} \frac{1}{x-3}(\frac{1}{\sqrt{x+1}}-\frac{1}{2})\)

Calculate the following limits analytically. \(\lim _{t \rightarrow 1/3} \frac{t-1/3}{3t-1^2}\)

Calculate the following limits analytically. \(\lim _{x \rightarrow 3} \frac{x^4-81}{x-3}\)

Calculate the following limits analytically. \(\lim _{p \rightarrow 1} \frac{p^5-1}{p-1}\)

Calculate the following limits analytically. \(\lim _{x \rightarrow 81} \frac{^4\sqrt x-3}{x-81}\)

Calculate the following limits analytically. \(\lim _{\theta \rightarrow \pi/4} \frac{\sin^2 \theta - \cos^2 \theta}{\sin \theta - \cos \theta}\)

Calculate the following limits analytically. \(\lim _{x \rightarrow \pi/2} \frac{\frac{1}{\sqrt{\sin x}}-1}{x+\pi/2}\)

Evaluate \(\lim _{x \rightarrow 1^+} \sqrt{\frac{x-1}{x-3}}\) and \(\lim _{x \rightarrow 1^-} \sqrt{\frac{x-1}{x-3}}\)

Applying the Squeeze Theorem a. Use graphing utility to illustrate the inequalities \(\cos x\ \leq\ \frac{\sin x}{x}\ \leq\ \frac{1}{\cos x}\) on [-1, 1]. b. Use part (a) and the Squeeze Theorem to explain why \(\lim _{x \rightarrow 0} \frac{\sin x}{x}=1\)

Assume the function g satisfies the inequality \(1\ \leq\ g(x)\ \leq\ sin^2x+1\) for x near 0. Use the Squeeze Theorem to find \(\lim _{x \rightarrow 0} g(x)\).

Determine the following infinite limits, if possible. \(\lim _{x \rightarrow 5} \frac{x-7}{x(x-5)^2}\)

Determine the following infinite limits, if possible. \(\lim _{x \rightarrow 5^+} \frac{x-5}{x+5}\)

Determine the following infinite limits, if possible. \(\lim _{x \rightarrow 3^-} \frac{x-4}{x^2-3x}\)

Determine the following infinite limits, if possible. \(\lim _{u \rightarrow 0^+} \frac{u-1}{\sin u}\)

Determine the following infinite limits, if possible. \(\lim _{x \rightarrow 0^-} \frac{2}{\tan x}\)

Let \(f(x)=\frac{x^2-5x+6}{x^2 -2x}\). a. Calculate \(\lim _{x \rightarrow 0^-} f(x),\ \lim _{x \rightarrow 0^+} f(x),\ \lim _{x \rightarrow 2^-} f(x)\), and \(\lim _{x \rightarrow 2^+} f(x)\). b. Does the graph of f have any vertical asymptotes? Explain. c. Graph f and then sketch the graph with paper and pencil, correcting any errors obtained with the graphing utility.

Evaluate the following limits. \(\lim _{x \rightarrow \infty} \frac{2x-3}{4x+10}\)

Evaluate the following limits. \(\lim _{x \rightarrow \infty} \frac{x^4-1}{x^5+2}\)

Evaluate the following limits. \(\lim _{x \rightarrow -\infty} (-3x^3+5)\)

Evaluate the following limits. \(\lim _{x \rightarrow \infty} (e^{-2z}+\frac{2}{z})\)

Evaluate the following limits. \(\lim _{x \rightarrow \infty} (3 \tan^{-1}x+2)\)

Evaluate the following limits. \(\lim _{x \rightarrow \infty} \frac{1}{\ln\ r+1}\)

Determine the end behavior of the following functions. \(f(x)=\frac{4x^3+1}{1-x^3}\)

Determine the end behavior of the following functions. \(f(x)=\frac{x+1}{\sqrt{9x^2+x}}\)

Determine the end behavior of the following functions. \(f(x)=1-e^{-2x}\)

Determine the end behavior of the following functions. \(f(x)=\frac{1}{\ln\ x^2}\)

Find all vertical and horizontal asymptotes of the following functions. \(f(x)=\frac{1}{\tan^{-1}x}\)

Find all vertical and horizontal asymptotes of the following functions. \(f(x)=\frac{2x^2+6}{2x^2+3x-2}\)

Determine whether the following functions are continuous at x = a using the continuity checklist to justify your answers. \(f(x)=\frac{1}{x-5};\ \ \ a=5\)

Determine whether the following functions are continuous at x = a using the continuity checklist to justify your answers. \(g(x)=\left\{\begin{array}{ll} \frac{x^{2}-16}{x-4} & \text { if } x \neq 4 \\ 9 & \text { if } x=4 \end{array} ; a=4\right.\)

Determine whether the following functions are continuous at x = a using the continuity checklist to justify your answers. \(h(x)=\sqrt{x^2-9};\ \ \ a=3\)

Determine whether the following functions are continuous at x = a using the continuity checklist to justify your answers. \(g(x)=\left\{\begin{array}{ll} \frac{x^{2}-16}{x-4} & \text { if } x \neq 4 \\ 8 & \text { if } x=4 \end{array} ; a=4\right.\)

Find the intervals on which the following functions are continuous. Specify right or left continuity at the endpoints. \(f(x)=\sqrt{x^2-5}\)

Find the intervals on which the following functions are continuous. Specify right or left continuity at the endpoints. \(g(x)=e^{\sqrt{x-2}}\)

Find the intervals on which the following functions are continuous. Specify right or left continuity at the endpoints. \(h(x)=\frac{2x}{x^3-25x}\)

Find the intervals on which the following functions are continuous. Specify right or left continuity at the endpoints. \(g(x)=\cos (e^x)\)

Let \(g(x)=\left\{\begin{array}{ll} 5 x-2 & \text { if } x<1 \\ a & \text { if } x=1 \\ a x^{2}+b x & \text { if } x>1 \end{array}\right.\) Determine values of the constants a and b for which g is continuous at x = 1.

Left and right continuity a. Is \(h(x)=\sqrt{x^2-9}\) left-continuous at x = 3? Explain. b. Is \(h(x)=\sqrt{x^2-9}\) right-continuous at x = 3? Explain.

Sketch the graph of a function that is continuous on (0, 1] and continuous on (1, 2) but is not continuous on (0, 2).

Intermediate Value Theorem a. Use the Intermediate Value Theorem to show that the equation \(x^5+7x+5=0\) has a solution in the interval (-1, 0). b. Find a solution to \(x^5+7x+5=0\) in (-1, 0) using a root finder.

The amount of an antibiotic (in mg) in the blood t hours after an intravenous line is opened is given by \(m(t)=100(e^{-0.1t}-e^{-0.3t})\). a. Use the Intermediate Value Theorem to show the amount of drug is 30 mg at some time in the interval [0, 5] and again at some time in the interval [5, 15]. b. Estimate the times at which m = 30 mg. c. Is the amount of drug in the blood ever 50 mg?

Give a formal proof that \(\lim _{x \rightarrow 1} (5x-2)=3\).

Give a formal proof that \(\lim _{x \rightarrow 5} \frac{x^2-25}{x-5}=10\).

Limit proofs a. Assume \(|f(x)|\ \leq\ L\) for all x near a and \(\lim _{x \rightarrow a} g(x)=0\). Give a formal proof that \(\lim _{x \rightarrow a}\ [f(x)g(x)]=0\). b. Find a function f for which \(\lim _{x \rightarrow 2}\ [f(x)(x-2)]\ \neq\ 0\). Why doesn’t this violate the result stated in (a)? c. The Heaviside function is defined as \(H(x)=\left\{\begin{array}{ll} 0 & \text { if } x<0 \\ 1 & \text { if } x \geq 0 \end{array}\right.\) Explain why \(\lim _{x \rightarrow 0}\ [xH(x)]=0\).

Give a formal proof that \(\lim _{x \rightarrow 2}\ \frac{1}{(x-2)^4}=\infty\).