?Suppose you want to estimate \(\lim \limits_{x \rightarrow a} \frac{f(x)}{g(x)}\). If

Explain in words the meaning of \(\lim \limits_{x \rightarrow a} f(x)=L\).

True or false: When \(\lim \limits_{x \rightarrow a} f(x)\) exists, it always equals f(a). Explain.

Explain the meaning of \(\lim \limits_{x \rightarrow a^{+}} f(x)=L\).

Explain the meaning of \(\lim \limits_{x \rightarrow a^{-}} f(x)=L\).

If \(\lim \limits_{x \rightarrow a^{-}} f(x)=L\) and \(\lim \limits_{x \rightarrow a^{+}} f(x)=M\), where L and M are finite real numbers, then what must be true about L and M in order for \(\lim \limits_{x \rightarrow a} f(x)\) to exist?

What are the potential problems of using a graphing utility to determine \(\lim \limits_{x \rightarrow a} f(x)\)?

Use the graph of h in the figure to find the following values, if they exist. a. h(2) b. \(\lim \limits_{x \rightarrow 2} h(x)\) c. h(4) d. \(\lim \limits_{x \rightarrow 4} h(x)\) d. \(\lim \limits_{x \rightarrow 5} h(x)\)

Use the graph of g in the figure to find the following values, if they exist. a. g(0) b. \(\lim \limits_{x \rightarrow 0} g(x)\) c. g(1) d. \(\lim \limits_{x \rightarrow 1} g(x)\)

Use the graph of f in the figure to find the following values, if they exist. a. f(1) b. \(\lim \limits_{x \rightarrow 1} f(x)\) c. f(0) d. \(\lim \limits_{x \rightarrow 0} f(x)\)

Use the graph of f in the figure to find the following values, if they exist. a. f(2) b. \(\lim \limits_{x \rightarrow 2} f(x)\) c. \(\lim \limits_{x \rightarrow 4} f(x)\) d. \(\lim \limits_{x \rightarrow 5} f(x)\)

Let \(f(x)=\frac{x^{2}-4}{x-2}\). a. Calculate f(x) for each value of x in the following table. b. Make a conjecture about the value of \(\lim \limits_{x \rightarrow 2} \frac{x^{2}-4}{x-2}\).

Let \(f(x)=\frac{x^{3}-1}{x-1}\). a. Calculate f(x) for each value of x in the following table. b. Make a conjecture about the value of \(\lim \limits_{x \rightarrow 1} \frac{x^{3}-1}{x-1}\).

Let \(g(t)=\frac{t-9}{\sqrt{t}-3}\). a. Make two tables, one showing the values of g for t = 8.9, 8.99, and 8.999 and one showing values of g for t = 9.1,9.01, and 9.001. b. Make a conjecture about the value of \(\lim \limits_{t \rightarrow 9} \frac{t-9}{\sqrt{t}-3}\).

Let \(f(x)=(1+x)^{1 / x}\). a. Make two tables, one showing the values of f for x = 0.01, 0.001, 0.0001, and 0.00001 and one showing values of f for x = -0.01, -0.001, -0.0001, and -0.00001. Round your answers to five digits. b. Estimate the value of \(\lim \limits_{x \rightarrow 0}(1+x)^{1 / x}\). c. What mathematical constant does \(\lim \limits_{x \rightarrow 0}(1+x)^{1 / x}\) appear to equal?

Let \(f(x)=\frac{x^{2}-25}{x-5}\). Use tables and graphs to make a conjecture about the values of \(\lim \limits_{x \rightarrow 5^{+}} f(x)\), \(\lim \limits_{x \rightarrow 5^{-}} f(x)\), and \(\lim \limits_{x \rightarrow 5} f(x)\) if they exist.

Let \(g(x)=\frac{x-100}{\sqrt{x}-10}\). Use tables and graphs to make a conjecture about the values \(\lim \limits_{x \rightarrow 100^{+}} g(x)\), \(\lim \limits_{x \rightarrow 100^{-}} g(x)\) and \(\lim \limits_{x \rightarrow 100} g(x)\), if they exist.

Use the graph of f in the figure to find the following values, if they exist. If a limit does not exist, explain why. a. f(1) b. \(\lim \limits_{x \rightarrow 1^{-}} f(x)\) c. \(\lim \limits_{x \rightarrow 1^{+}} f(x)\) d. \(\lim \limits_{x \rightarrow 1} f(x)\)

Use the graph of g in the figure to find the following values, if they exist. If a limit does not exist, explain why. a. g(2) b. \(\lim \limits_{x \rightarrow 2^{-}} g(x)\) c. \(\lim \limits_{x \rightarrow 2^{+}} g(x)\) d. \(\lim \limits_{x \rightarrow 2} g(x)\) e. g(3) f. \(\lim \limits_{x \rightarrow 3^{-}} g(x)\) g. \(\lim \limits_{x \rightarrow 3^{+}} g(x)\) h. g(4) i. \(\lim \limits_{x \rightarrow 4} g(x)\)

Use the graph of f in the figure to find the following values, if they exist. If a limit does not exist, explain why. a. f(1) b. \(\lim \limits_{x \rightarrow 1^{-}} f(x)\) c. \(\lim \limits_{x \rightarrow 1^{+}} f(x)\) d. \(\lim \limits_{x \rightarrow 1} f(x)\) e. f(3) f. \(\lim \limits_{x \rightarrow 3^{-}} f(x)\) g. \(\lim \limits_{x \rightarrow 3^{+}} f(x)\) h. \(\lim \limits_{x \rightarrow 3} f(x)\) i. f(2) j. \(\lim \limits_{x \rightarrow 2^{-}} f(x)\) k. \(\lim \limits_{x \rightarrow 2^{+}} f(x)\) l. \(\lim \limits_{x \rightarrow 2} f(x)\)

Use the graph of g in the figure on the next page to find the following values, if they exist. If a limit does not exist, explain why. a. g(-1) b. \(\lim \limits_{x \rightarrow-1^{-}} g(x)\) c. \(\lim \limits_{x \rightarrow-1^{+}} g(x)\) d. \(\lim \limits_{x \rightarrow-1} g(x)\) e. g(1) f. \(\lim \limits_{x \rightarrow 1} g(x)\) g. \(\lim \limits_{x \rightarrow 3} g(x)\) h. g(5) i. \(\lim \limits_{x \rightarrow 5^{-}} g(x)\)

Strange behavior near x = 0 a. Create a table of values of \(\sin \left(\frac{1}{x}\right)\) for \(x=\frac{2}{\pi}, \frac{2}{3 \pi}, \frac{2}{5 \pi}, \frac{2}{7 \pi}, \frac{2}{9 \pi}\) and \(\frac{2}{11 \pi}\). Describe the pattern of values you observe. b. Why does a graphing utility have difficulty plotting the graph of y = sin (1/x) near x = 0 (see figure)? c. What do you conclude about \(\lim \limits_{x \rightarrow 0} \sin (1 / x)\)?

Strange behavior near r = 0 a. Create a table of values of tan (3/x) for x = 1, 0.1, 0.01, 0.001, 0.0001, and 0.00001. Describe the general pattern in the values you observe. b. Use a graphing utility to graph y = tan (3/x). Why does a graphing utility have difficulty plotting the graph near x = 0? c. What do you conclude about \(\lim \limits_{x \rightarrow 0} \tan (3 / x)\)?

Determine whether the following statements are true and give an explanation or counterexample. a. The value of \(\lim \limits_{x \rightarrow 3} \frac{x^{2}-9}{x-3}\) does not exist. b. The value of \(\lim \limits_{x \rightarrow a} f(x)\) can always be found by computing f(a). c. The value of \(\lim \limits_{x \rightarrow a} f(x)\) does not exist if f(a) is undefined.

Sketch the graph of a function with the given properties. You do not need to find a formula for the function. f(1) = 0, f(2) = 4, f(3) = 6, \(\lim \limits_{x \rightarrow 2^{-}} f(x)=-3\), \(\lim \limits_{x \rightarrow 2^{+}} f(x)=5\)

Sketch the graph of a function with the given properties. You do not need to find a formula for the function. g(1) = 0, g(2) = 1, g(3) = -2, \(\lim \limits_{x \rightarrow 2} g(x)=0\), \(\lim \limits_{x \rightarrow 3^{-}} g(x)=-1, \lim \limits_{x \rightarrow 3^{+}} g(x)=-2\)

Estimate the value of the following limits by creating a table of function values for h = 0.01, 0.001, and 0.0001, and h = -0.01, -0.001, and -0.0001. \(\lim \limits_{h \rightarrow 0}(1+2 h)^{1 / h}\)

Estimate the value of the following limits by creating a table of function values for h = 0.01, 0.001, and 0.0001, and h = -0.01, -0.001, and -0.0001. \(\lim \limits_{h \rightarrow 0}(1+3 h)^{2 / h}\)

Estimate the value of the following limits by creating a table of function values for h = 0.01, 0.001, and 0.0001, and h = -0.01, -0.001, and -0.0001. \(\lim \limits_{h \rightarrow 0} \frac{2^{h}-1}{h}\)

Estimate the value of the following limits by creating a table of function values for h = 0.01, 0.001, and 0.0001, and h = -0.01, -0.001, and -0.0001. \(\lim \limits_{h \rightarrow 0} \frac{\ln (1+h)}{h}\)

Let \(f(x)=\frac{|x|}{x}\) for \(x \neq 0\). a. Sketch a graph of f on the interval [-2,2]. b. Does \(\lim \limits_{x \rightarrow 0} f(x)\) exist? Explain your reasoning after first examining \(\lim \limits_{x \rightarrow 0^{-}} f(x)\) and \(\lim \limits_{x \rightarrow 0^{+}} f(x)\).

For any real number x, the floor function (or greatest integer function), \(\lfloor x\rfloor\), is defined to be the greatest integer less than or equal to x (see figure). a. Compute \(\lim \limits_{x \rightarrow-1^{-}}\lfloor x\rfloor, \lim \limits_{x \rightarrow-1^{+}}\lfloor x\rfloor, \lim \limits_{x \rightarrow 2^{-}}\lfloor x\rfloor \text {, and } \lim \limits_{x \rightarrow 2^{+}}\lfloor x\rfloor\). b. Compute \(\lim \limits_{x \rightarrow 2.3^{-}}\lfloor x\rfloor, \lim \limits_{x \rightarrow 2.3^{+}}\lfloor x\rfloor, \text { and } \lim \limits_{x \rightarrow 2.3}\lfloor x\rfloor\). c. In general, for an integer a, state the values of \(\lim \limits_{x \rightarrow a^{-}}\lfloor x\rfloor\) and \(\lim \limits_{x \rightarrow a^{+}}\lfloor x\rfloor\). d. In general, if a is not an integer, state the values of \(\lim \limits_{x \rightarrow a^{-}}\lfloor x\rfloor\) and \(\lim \limits_{x \rightarrow a^{+}}\lfloor x\rfloor\). e. For what values of a does \(\lim \limits_{x \rightarrow a}\lfloor x\rfloor\) exist? Explain.

The ceiling function For any real number x, the ceiling function, \(\lceil x\rceil\), is defined to be the least integer greater than or equal to x. a. Graph the ceiling function \(y=\lceil x\rceil\) for \(-2 \leq x \leq 3\). b. Evaluate \(\lim \limits_{x \rightarrow 2^{-}}\lceil x\rceil, \lim \limits_{x \rightarrow 1^{+}}\lceil x\rceil, \text { and } \lim \limits_{x \rightarrow 1.5}\lceil x\rceil\). c. For what values of a does \(\lim \limits_{x \rightarrow a}\lceil x\rceil\) exist? Explain.

Use the zoom and trace features of a graphing utility to approximate \(\lim \limits_{x \rightarrow 1} \frac{\sqrt{2 x-x^{4}}-\sqrt[3]{x}}{1-x^{3 / 4}}\).

Use the zoom and trace features of a graphing utility to approximate \(\lim \limits_{x \rightarrow 0} \frac{6^{x}-3^{x}}{2 x}\).

Assume that postage for sending a first-class letter in the United States is $0.42 for the first ounce (up to and including 1 oz) plus $0.17 for each additional ounce (up to and including each additional ounce). a. Graph the function p = f(w) that gives the postage p for sending a letter that weighs w ounces if \(0<w \leq 5\). b. Evaluate \(\lim \limits_{w \rightarrow 3.3} f(w)\). c. Interpret the limits \(\lim \limits_{w \rightarrow 1^{+}} f(w) \text { and } \lim \limits_{w \rightarrow 1^{-}} f(w)\). d. Does \(\lim \limits_{w \rightarrow 4} f(w)\) exist? Explain.

The Heaviside function is used in engineering applications to simulate flipping a switch. It is defined as \(H(x)=\left\{\begin{array}{ll} 0 & \text { if } x<0 \\ 1 & \text { if } x \geq 0 \end{array}\right.\) a. Sketch a graph of H on the interval [ -1, 2]. b. Does \(\lim \limits_{x \rightarrow 0} H(x)\) exist? Explain your reasoning after first examining \(\lim \limits_{x \rightarrow 0^{-}} H(x) \text { and } \lim \limits_{x \rightarrow 0^{+}} H(x)\).

Suppose you want to estimate \(\lim \limits_{x \rightarrow a} \frac{f(x)}{g(x)}\). If g(x) is nearly equal to 0 when x is close to a, then a calculator utility may round the value of g(x) to 0. Evaluate \(\frac{\sin x^{20}}{x^{20}}\) for x = 0.1, 0.01,...,0.00001. Based on the value given by your calculator, propose a value for \(\lim \limits_{x \rightarrow 0^{+}} \frac{\sin x^{20}}{x^{20}}\). Using limit techniques introduced later in the text, it can be shown that \(\lim \limits_{x \rightarrow 0^{+}} \frac{\sin x^{20}}{x^{20}}=1\). Does your proposed value for the limit agree? Explain.

Suppose you want to estimate \(\lim \limits_{x \rightarrow a} \frac{f(x)}{g(x)}\). If g(x) is nearly equal to 0 when x is close to a, then a calculator utility may round the value of g(x) to 0. a. Graph the function \(y=\frac{x \sin x}{1-\cos x}\) with the window \([-1,1] \times[0,3]\). Based upon this graph, estimate the value of \(\lim \limits_{x \rightarrow 0} \frac{x \sin x}{1-\cos x}\). What is the value of \(\frac{x \sin x}{1-\cos x} \text { at } x=0\)? b. Values of \(\frac{x \sin x}{1-\cos x}\) near 0, computed with a computer algebra system, are shown in the accompanying table. It appears that \(\lim \limits_{x \rightarrow 0} \frac{x \sin x}{1-\cos x}\) is undefined (does not exist). Using limit techniques introduced later in the text, it can be shown that \(\lim \limits_{x \rightarrow 0} \frac{x \sin x}{1-\cos x}=2\). What do you think happened, and why?

A function f is even if f(-x) = f(x) for all x in the domain of f. If f is even, with \(\lim \limits_{x \rightarrow 2^{+}} f(x)=5\) and \(\lim \limits_{x \rightarrow 2^{-}} f(x)=8\), find the following limits. a. \(\lim \limits_{x \rightarrow-2^{+}} f(x)\) b. \(\lim \limits_{x \rightarrow-2^{-}} f(x)\)

A function g is odd if g(-x) = -g(x) for all x in the domain of g. If g is odd, with \(\lim \limits_{x \rightarrow 2^{+}} g(x)=5\) and \(\lim \limits_{x \rightarrow 2^{-}} g(x)=8\), find the following limits. a. \(\lim \limits_{x \rightarrow-2^{+}} g(x)\) b. \(\lim \limits_{x \rightarrow-2^{-}} g(x)\)

a. Use a graphing utility to estimate the values of \(\lim \limits_{x \rightarrow 0} \frac{\tan 2 x}{\sin x}\), \(\lim \limits_{x \rightarrow 0} \frac{\tan 3 x}{\sin x} \text { and } \lim \limits_{x \rightarrow 0} \frac{\tan 4 x}{\sin x}\). b. Make a conjecture about the value of \(\lim \limits_{x \rightarrow 0} \frac{\tan n x}{\sin x}\) for any real constant n.

Graph \(f(x)=\frac{\sin n x}{x}\) for n = 1, 2, 3, and 4 (four graphs). Use the window \([-1,1] \times[0,5]\). a. Estimate the values of \(\lim \limits_{x \rightarrow 0} \frac{\sin x}{x}, \lim \limits_{x \rightarrow 0} \frac{\sin 2 x}{x}, \lim \limits_{x \rightarrow 0} \frac{\sin 3 x}{x}\), and \(\lim \limits_{x \rightarrow 0} \frac{\sin 4 x}{x}\). b. Make a conjecture about the value of \(\lim \limits_{x \rightarrow 0} \frac{\sin n x}{x}\) for any real constant n.

Use a graphing utility to plot \(y=\frac{\sin n x}{\sin m x}\) for at least three different pairs of nonzero constants m and n of your choice. Estimate \(\lim \limits_{x \rightarrow 0} \frac{\sin n x}{\sin m x}\) in each case. Then use your work to make a conjecture about the value of \(\lim \limits_{x \rightarrow 0} \frac{\sin n x}{\sin m x}\) for any nonzero values of m and n.