Chapter 2

Evaluate the following limits. $$\lim _{\theta \rightarrow \infty} \frac{\cos \theta}{\theta^{2}}$$

Evaluate the following limits. \(\lim _{x \rightarrow 4}(3 x-7)\)

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 \(\lim _{x \rightarrow 1} \frac{x^{3}-1}{x-1}\)

Evaluate the following limits. $$\lim _{x \rightarrow \infty} \frac{3+2 x+4 x^{2}}{x^{2}}$$

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

The table gives the position \(s(t)\) of an object moving along a line at time \(t,\) over a two-second interval. Find the average velocity of the object over the following intervals. a. [0,2] b. [0,1.5] c. [0,1] d. [0,0.5]

Let \(g(t)=\frac{t-9}{\sqrt{t}-3}\) a. Make two tables, one showing 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 \(\lim _{t \rightarrow 9} \frac{t-9}{\sqrt{t}-3}\)

Evaluate the following limits. $$\lim _{x \rightarrow \infty} \frac{\cos x^{5}}{\sqrt{x}}$$

Let \(f(x)=x^{3}+3\) and note that \(\lim _{x \rightarrow 0} f(x)=3\) For each value of \(\varepsilon,\) use a graphing utility to find all values of \(\delta>0\) such that \(|f(x)-3|<\varepsilon\) whenever \(0<|x-0|<\delta .\) Sketch graphs illustrating your work. a. \(\varepsilon=1\) b. \(\varepsilon=0.5\)

Graph the function \(f(x)=\frac{1}{x^{2}-x}\) using a graphing utility with the window \([-1,2] \times[-10,10] .\) Use your graph to determine the following limits. $$\text { a. } \lim _{x \rightarrow 0^{-}} f(x)$$ $$\text { b. } \lim _{x \rightarrow 0^{+}} f(x)$$ $$\text { c. } \lim _{x \rightarrow 1^{-}} f(x)$$ $$\text { d. } \lim _{x \rightarrow 1^{+}} f(x)$$

Evaluate the following limits. \(\lim _{x \rightarrow-9} 5 x\)

Consider the position function \(s(t)=-16 t^{2}+100 t\) representing the position of an object moving vertically along a line. Sketch a graph of \(s\) with the secant line passing through \((0.5, s(0.5))\) and \((2, s(2)) .\) Determine the slope of the secant line and explain its relationship to the moving object.

Continuity at a point Determine whether the following functions are continuous at a. Use the continuity checklist to justify your answer. $$f(x)=\frac{2 x^{2}+3 x+1}{x^{2}+5 x} ; a=5$$

Let \(f(x)=(1+x)^{1 / x}\) a. Make two tables, one showing 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 \(\lim _{x \rightarrow 0}(1+x)^{1 / x}\) c. What mathematical constant does \(\lim _{x \rightarrow 0}(1+x)^{1 / x}\) appear to equal?

Evaluate the following limits. $$\lim _{x \rightarrow-\infty}\left(5+\frac{100}{x}+\frac{\sin ^{4} x^{3}}{x^{2}}\right)$$

Graph the function \(f(x)=\frac{e^{-x}}{x(x+2)^{2}}\) using a graphing utility. (Experiment with your choice of a graphing window.) Use your graph to determine the following limits. $$\text { a. } \lim _{x \rightarrow-2^{+}} f(x)$$ $$\text { b. } \lim _{x \rightarrow-2} f(x)$$ $$\text { c. } \lim _{x \rightarrow 0^{-}} f(x)$$ $$\text { d. } \lim _{x \rightarrow 0^{+}} f(x)$$

Let \(g(x)=2 x^{3}-12 x^{2}+26 x+4\) and note that \(\lim _{x \rightarrow 2} g(x)=24\) For each value of \(\varepsilon,\) use a graphing utility to find all values of \(\delta>0\) such that \(|g(x)-24|<\varepsilon\) whenever \(0<|x-2|<\delta\) Sketch graphs illustrating your work. a. \(\varepsilon=1\) b. \(\varepsilon=0.5\)

Evaluate the following limits. \(\lim _{x \rightarrow 2}(-3 x)\)

Consider the position function \(s(t)=\sin \pi t\) representing the position of an object moving along a line on the end of a spring. Sketch a graph of \(s\) together with a secant line passing through \((0, s(0))\) and \((0.5, s(0.5)) .\) Determine the slope of the secant line and explain its relationship to the moving object.

Continuity at a point Determine whether the following functions are continuous at a. Use the continuity checklist to justify your answer. $$f(x)=\frac{2 x^{2}+3 x+1}{x^{2}+5 x} ; a=-5$$

Let \(f(x)=\frac{x-2}{\ln |x-2|}\) a. Graph \(f\) to estimate \(\lim _{x \rightarrow 2} f(x)\) b. Evaluate \(f(x)\) for values of \(x\) near 2 to support your conjecture in part (a).

Evaluate the following limits. \(\lim _{x \rightarrow 6} 4\)

Consider the position function \(s(t)=-16 t^{2}+128 t\) (Exercise 9). Complete the following table with the appropriate average velocities. Then make a conjecture about the value of the instantaneous velocity at \(t=1\) $$\begin{array}{|l|l|l|l|l|l|}\hline \begin{array}{l}\text { Time } \\\\\text { interval }\end{array} & {[1,2]} & {[1,1.5]} & {[1,1.1]} & {[1,1.01]} & {[1,1.001]} \\\\\hline \begin{array}{l}\text { Average } \\\\\text { velocity }\end{array} & & & & & \\\\\hline\end{array}$$

Continuity at a point Determine whether the following functions are continuous at a. Use the continuity checklist to justify your answer. $$f(x)=\sqrt{x-2} ; a=1$$

Let \(g(x)=\frac{e^{2 x}-2 x-1}{x^{2}}\) a. Graph \(g\) to estimate \(\lim _{x \rightarrow 0} g(x)\) b. Evaluate \(g(x)\) for values of \(x\) near 0 to support your conjecture in part (a).

Determine the following limits. $$\lim _{x \rightarrow-\infty} 3 x^{11}$$

The function \(f\) in the figure satisfies \(\lim _{x \rightarrow 4} f(x)=5 .\) For each value of \(\varepsilon,\) find all values of \(\delta>0\) such that $$|f(x)-5|<\varepsilon \quad \text { whenever } \quad 0<|x-4|<\delta\quad(3)$$ a. \(\varepsilon=2\) b. \(\varepsilon=1\) c. For any \(\varepsilon>0,\) make a conjecture about the corresponding value of \(\delta\) satisfying (3)

Consider the position function \(s(t)=-4.9 t^{2}+30 t+20\) (Exercise 10). Complete the following table with the appropriate average velocities. Then make a conjecture about the value of the instantaneous velocity at \(t=2\). $$\begin{array}{|l|l|l|l|l|l|}\hline \begin{array}{l}\text { Time } \\\\\text { interval }\end{array} & {[2,3]} & {[2,2.5]} & {[2,2.1]} & {[2,2.01]} & {[2,2.001]} \\\\\hline \begin{array}{l} \text { Average } \\\\\text { velocity }\end{array} & & & & & \\\\\hline\end{array}$$

Let \(f(x)=\frac{1-\cos (2 x-2)}{(x-1)^{2}}\) a. Graph \(f\) to estimate \(\lim _{x \rightarrow 1} f(x)\) b. Evaluate \(f(x)\) for values of \(x\) near 1 to support your conjecture in part (a).

Determine the following limits. $$\lim _{x \rightarrow \infty} x^{-6}$$

Determining limits analytically Determine the following limits or state that they do not exist. $$\lim _{x \rightarrow 2^{+}} \frac{1}{x-2}$$ $$\text { b. } \lim _{x \rightarrow 2^{-}} \frac{1}{x-2}$$ $$\text { c. } \lim _{x \rightarrow 2} \frac{1}{x-2}$$

Assume \(\lim _{x \rightarrow 1} f(x)=8, \lim _{x \rightarrow 1} g(x)=3,\) and \(\lim _{x \rightarrow 1} h(x)=2 .\) Compute the following limits and state the limit laws used to justify your computations. \(\lim _{x \rightarrow 1}(4 f(x))\)

The following table gives the position \(s(t)\) of an object moving along a line at time \(t .\) Determine the average velocities over the time intervals \([1,1.01],[1,1.001]\) and [1,1.0001] . Then make a conjecture about the value of the instantaneous velocity at \(t=1.\) $$\begin{array}{|l|l|l|l|l|}\hline t & 1 & 1.0001 & 1.001 & 1.01 \\\\\hline s(t) & 64 & 64.00479984 & 64.047984 & 64.4784 \\\\\hline\end{array}$$

Continuity at a point Determine whether the following functions are continuous at a. Use the continuity checklist to justify your answer. $$f(x)=\left\\{\begin{array}{ll}\frac{x^{2}-1}{x-1} & \text { if } x \neq 1 \\\ 3 & \text { if } x=1\end{array} ; a=1\right.$$

Let \(g(x)=\frac{3 \sin x-2 \cos x+2}{x}\) a. Graph of \(g\) to estimate \(\lim _{x \rightarrow 0} g(x)\) b. Evaluate \(g(x)\) for values of \(x\) near 0 to support your conjecture in part (a).

Determining limits analytically Determine the following limits or state that they do not exist. a. \(\lim _{x \rightarrow 3^{+}} \frac{2}{(x-3)^{3}}\) b. \(\lim _{x \rightarrow 3^{-}} \frac{2}{(x-3)^{3}} \quad\) c. \(\lim _{x \rightarrow 3} \frac{2}{(x-3)^{3}}\)

Determine the following limits. $$\lim _{x \rightarrow-\infty} x^{-11}$$

Assume \(\lim _{x \rightarrow 1} f(x)=8, \lim _{x \rightarrow 1} g(x)=3,\) and \(\lim _{x \rightarrow 1} h(x)=2 .\) Compute the following limits and state the limit laws used to justify your computations. \(\lim _{x \rightarrow 1} \frac{f(x)}{h(x)}\)

The following table gives the position \(s(t)\) of an object moving along a line at time \(t .\) Determine the average velocities over the time intervals [2,2.01],[2,2.001] and \([2,2.0001] .\) Then make a conjecture about the value of the instantaneous velocity at \(t=2\) $$\begin{array}{|l|c|c|c|c|}\hline t & 2 & 2.0001 & 2.001 & 2.01 \\\\\hline s(t) & 56 & 55.99959984 & 55.995984 & 55.9584 \\\\\hline\end{array}$$

Continuity at a point Determine whether the following functions are continuous at a. Use the continuity checklist to justify your answer. $$f(x)=\left\\{\begin{array}{ll}\frac{x^{2}-4 x+3}{x-3} & \text { if } x \neq 3 \\\2 & \text { if } x=3\end{array} ; a=3\right.$$

Let \(f(x)=\frac{x^{2}-25}{x-5}\). Use tables and graphs to make a conjecture about the values of \(\lim _{x \rightarrow 5^{+}} f(x)\) \(\lim _{t \rightarrow 0} f(x),\) and \(\lim _{t \rightarrow 5} f(x)\) or state that they do not exist.

Determining limits analytically Determine the following limits or state that they do not exist. a. \(\lim _{x \rightarrow 4^{+}} \frac{x-5}{(x-4)^{2}}\) b. \(\lim _{x \rightarrow 4^{-}} \frac{x-5}{(x-4)^{2}} \quad\) c. \(\lim _{x \rightarrow 4} \frac{x-5}{(x-4)^{2}}\)

Determine the following limits. $$\lim _{x \rightarrow \infty}\left(3 x^{12}-9 x^{7}\right)$$

Use the precise definition of a limit to prove the following limits. $$\lim _{x \rightarrow 1}(8 x+5)=13$$

Assume \(\lim _{x \rightarrow 1} f(x)=8, \lim _{x \rightarrow 1} g(x)=3,\) and \(\lim _{x \rightarrow 1} h(x)=2 .\) Compute the following limits and state the limit laws used to justify your computations. \(\lim _{x \rightarrow 1}(f(x)-g(x))\)

Consider the position function \(s(t)=-16 t^{2}+100 t .\) Complete the following table with the appropriate average velocities. Then make a conjecture about the value of the instantaneous velocity at \(t=3\) $$\begin{array}{|l|l|}\hline \text { Time interval } & \text { Average velocity } \\\\\hline[2,3] & \\ \hline[2.9,3] & \\\\\hline[2.99,3] & \\\\\hline[2.999,3] & \\\\\hline[2.9999,3] & \\\\\hline\end{array}$$

Continuity at a point Determine whether the following functions are continuous at a. Use the continuity checklist to justify your answer. $$f(x)=\frac{5 x-2}{x^{2}-9 x+20} ; a=4$$

Determining limits analytically Determine the following limits or state that they do not exist. a. \(\lim _{x \rightarrow 1^{+}} \frac{x-2}{(x-1)^{3}}\) b. \(\lim _{x \rightarrow 1^{-}} \frac{x-2}{(x-1)^{3}} \quad\) c. \(\lim _{x \rightarrow 1} \frac{x-2}{(x-1)^{3}}\)

Determine the following limits. $$\lim _{x \rightarrow-\infty}\left(3 x^{7}+x^{2}\right)$$

Use the precise definition of a limit to prove the following limits. $$\lim _{x \rightarrow 3}(-2 x+8)=2$$