Chapter 2

How is \(\lim _{x \rightarrow a} p(x)\) calculated if \(p\) is a polynomial function?

Suppose \(x\) lies in the interval (1,3) with \(x \neq 2 .\) Find the smallest positive value of \(\delta\) such that the inequality \(0<|x-2|<\delta\) is true.

Which of the following functions are continuous for all values in their domain? Justify your answers. a. \(a(t)=\) altitude of a skydiver \(t\) seconds after jumping from a plane b. \(n(t)=\) number of quarters needed to park legally in a metered parking space for \(t\) minutes c. \(T(t)=\) temperature \(t\) minutes after midnight in Chicago on January 1 d. \(p(t)=\) number of points scored by a basketball player after \(t\) minutes of a basketball game

Explain the meaning of \(\lim _{x \rightarrow-\infty} f(x)=10\).

$$\text { Explain the meaning of } \lim _{n \rightarrow \infty} f(x)=-\infty$$

Suppose \(s(t)\) is the position of an object moving along a line at time \(t \geq 0 .\) What is the average velocity between the times \(t=a\) and \(t=b ?\)

$$\text { Explain the meaning of } \lim _{x \rightarrow a} f(x)=L$$.

Evaluate \(\lim _{x \rightarrow 1}\left(x^{3}+3 x^{2}-3 x+1\right)\).

$$\text { Explain the meaning of } \lim _{x \rightarrow a} f(x)=\infty$$

Give the three conditions that must be satisfied by a function to be continuous at a point.

Suppose \(s(t)\) is the position of an object moving along a line at time \(t\) as \(0 .\) Describe a process for finding the instantaneous velocity at \(t=a\)

For what values of \(a\) does \(\lim r(x)=r(a)\) if \(r\) is a rational function?

Which one of the following intervals is not symmetric about \(x=5 ?\) a. (1,9) b. (4,6)\(\quad\) c. (3,8) d. (4.5,5.5)

What does it mean for a function to be continuous on an interval?

The function \(s(t)\) represents the position of an object at time \(t\) moving along a line. Suppose \(s(2)=136\) and \(s(3)=156 .\) Find the average velocity of the object over the interval of time [2,3]

Evaluate \(\lim _{x \rightarrow 4}\left(\frac{x^{2}-4 x-1}{3 x-1}\right)\)

Suppose \(a\) is a constant and \(\delta\) is a positive constant. Give a geometric description of the sets \(\\{x:|x-a|<\delta\\}\) and \(\\{x: 0<|x-a|<\delta\\}\)

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

Consider the function \(F(x)=f(x) / g(x)\) with \(g(a)=0 .\) Does \(F\) necessarily have a vertical asymptote at \(x=a ?\) Explain your reasoning.

We informally describe a function \(f\) to be continuous at \(a\) if its graph contains no holes or breaks at \(a\). Explain why this is not an adequate definition of continuity.

The function \(s(t)\) represents the position of an object at time \(t\) moving along a line. Suppose \(s(1)=84\) and \(s(4)=144 .\) Find the average velocity of the object over the interval of time [1,4]

Explain why \(\lim _{x \rightarrow 3} \frac{x^{2}-7 x+12}{x-3}=\lim _{x \rightarrow 3}(x-4)\) and then evaluate \(\lim _{x \rightarrow 3} \frac{x^{2}-7 x+12}{x-3}\)

State the precise definition of \(\lim _{x \rightarrow a} f(x)=L\)

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

Evaluate \(\lim _{x \rightarrow 5}\left(\frac{4 x^{2}-100}{x-5}\right)\)

Interpret \(|f(x)-L|<\varepsilon\) in words.

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

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. $$\lim _{x \rightarrow 1}(4 f(x))$$

Determine the following limits at infinity. $$\lim _{t \rightarrow \infty}\left(-12 t^{-5}\right)$$

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 \(t=1\) $$\begin{array}{|l|l|l|l|l|} \hline t & 1 & 1.0001 & 1.001 & 1.01 \\ \hline s(t) & 64 & 64.00479984 & 64.047984 & 64.4784 \\ \hline \end{array}$$

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 \(\lim _{x \rightarrow 2} \frac{x^{2}-4}{x-2}\) $$\begin{array}{|l|l|l|l|l|}\hline x & 1.9 & 1.99 & 1.999 & 1.9999 \\\\\hline f(x)=\frac{x^{2}-4}{x-2} & & & & \\\\\hline x & 2.1 & 2.01 & 2.001 & 2.0001 \\\\\hline f(x)=\frac{x^{2}-4}{x-2} & & & & \\\\\hline\end{array}$$

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. $$\lim _{x \rightarrow 1} \frac{f(x)}{h(x)}$$

Determine the following limits at infinity. $$\lim _{x \rightarrow-\infty} 2 x^{-8}$$

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 instantancous velocity at \(t=2\) $$\begin{array}{|l|l|l|l|l|} \hline t & 2 & 2.0001 & 2.001 & 2.01 \\ \hline s(t) & 56 & 55.99959984 & 55.995984 & 55.9584 \\ \hline \end{array}$$

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}$$$\begin{array}{|l|l|l|l|l|}\hline x & 0.9 & 0.99 & 0.999 & 0.9999 \\\\\hline f(x)=\frac{x^{3}-1}{x-1} & & & & \\\\\hline x & 1.1 & 1.01 & 1.001 & 1.0001 \\\\\hline f(x)=\frac{x^{3}-1}{x-1} & & & & \\\\\hline\end{array}$$

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. $$\lim _{x \rightarrow 1}(f(x)-g(x))$$

Determine the following limits at infinity. $$\lim _{x \rightarrow \infty}\left(3+\frac{10}{x^{2}}\right)$$

Complete the following sentences in terms of a limit. a. A function is continuous from the left at \(a\) if ________. b. A function is continuous from the right at \(a\) if ________.

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 \(\lim _{t \rightarrow 9} \frac{t-9}{\sqrt{t}-3}\).

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. $$\lim _{x \rightarrow 1}(f(x) h(x))$$

Determine the following limits at infinity. $$\lim _{x \rightarrow \infty}\left(5+\frac{1}{x}+\frac{10}{x^{2}}\right)$$

Evaluate \(f(3)\) if \(\lim _{x \rightarrow 3^{-}} f(x)=5, \lim _{x \rightarrow 3^{+}} f(x)=6,\) and \(f\) is right-continuous at \(x=3\).

Describe a process for finding the slope of the line tangent to the graph of \(f\) at \((a, f(a))\)

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?

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. $$\lim _{x \rightarrow 1} \frac{f(x)}{g(x)-h(x)}$$

Determine the following limits at infinity. $$\lim _{x \rightarrow \infty} \frac{f(x)}{g(x)} \text { if } f(x) \rightarrow 100,000 \text { and } g(x) \rightarrow \infty \text { as } x \rightarrow \infty$$

Describe the parallels between finding the instantaneous velocity of an object at a point in time and finding the slope of the line tangent to the graph of a function at a point on the graph.

Explain the meaning of \(\lim _{x \rightarrow 0^{-}} f(x)=L\).

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. $$\lim _{x \rightarrow 1} \sqrt[3]{f(x) g(x)+3}$$

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