Chapter 2

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

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

How is \(\lim _{x \rightarrow a} f(x)\) calculated if \(f\) 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.

Use a graph to explain the meaning of \(\lim _{x \rightarrow a^{+}} f(x)=-\infty\)

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

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 \(t=b ?\)

What is a horizontal asymptote?

How are \(\lim _{x \rightarrow a^{+}} f(x)\) and \(\lim _{x \rightarrow a^{+}} f(x)\) calculated if \(f\) is a polynomial function?

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 \(t \geq 0 .\) Describe a process for finding the instantaneous velocity at \(t=a\).

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

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) c. (3,8) d. (4.5,5.5)

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

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

Assume \(\lim _{x \rightarrow 3} g(x)=4\) and \(f(x)=g(x)\) whenever \(x \neq 3 .\) Evalu\(\lim _{x \rightarrow 3} f(x),\) if possible.

Does the set \(\\{x: 0<|x-a|<\delta\\}\) include the point \(x=a ?\) Explain.

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.

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

Describe the end behavior of \(f(x)=-2 x^{3}\)

If \(\lim _{x \rightarrow a} f(x)=L\) and \(\lim _{x \rightarrow a^{+}} f(x)=M,\) where \(L\) and \(M\) are finite real numbers, then how are \(L\) and \(M\) related if \(\lim _{x \rightarrow a} f(x)\) exists?

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

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

Complete the following sentences. a. A function is continuous from the left at \(a\) if ______. b. A function is continuous from the right at \(a\) if ______.

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.

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

If \(\lim _{x \rightarrow 2} f(x)=-8,\) find \(\lim _{x \rightarrow 2}(f(x))^{2 / 3}\).

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

$$\text { Evaluate } \lim _{x \rightarrow 3^{-}} \frac{1}{x-3} \text { and } \lim _{x \rightarrow 3^{+}} \frac{1}{x-3}$$

Describe the points (if any) at which a rational function fails to be continuous.

Graph the parabola \(f(x)=x^{2} .\) Explain why the secant lines between the points \((-a, f(-a))\) and \((a, f(a))\) have zero slope. What is the slope of the tangent line at \(x=0 ?\)

Evaluate \(\lim _{x \rightarrow \infty} e^{x}, \lim _{x \rightarrow-\infty} e^{x},\) and \(\lim _{x \rightarrow \infty} e^{-x}\)

Suppose \(p\) and \(q\) are polynomials. If \(\lim _{x \rightarrow 0} \frac{p(x)}{q(x)}=10\) and \(q(0)=2,\) find \(p(0)\).

Analyzing infinite limits numerically Compute the values of \(f(x)=\frac{x+1}{(x-1)^{2}}\) in the following table and use them to determine \(\lim _{x \rightarrow 1} f(x)\) $$\begin{array}{|c|c|c|c|} \hline x & \frac{x+1}{(x-1)^{2}} & x & \frac{x+1}{(x-1)^{2}} \\ \hline 1.1 & & 0.9 & \\ \hline 1.01 & & 0.99 & \\ \hline 1.001 & & 0.999 & \\ \hline 1.0001 & & 0.9999 & \\ \hline \end{array}$$

What is the domain of \(f(x)=e^{x} / x\) and where is \(f\) continuous?

Average velocity 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 \([2,3].\)

Use a sketch to find the end behavior of \(f(x)=\ln x\)

Suppose \(\lim _{x \rightarrow 2} f(x)=\lim _{x \rightarrow 2} h(x)=5 .\) Find \(\lim _{x \rightarrow 2} g(x),\) where \(f(x) \leq g(x) \leq h(x),\) for all \(x\).

Use the graph of $$f(x)=\frac{x}{\left(x^{2}-2 x-3\right)^{2}} \text { to determine } \lim _{x \rightarrow-1} f(x) \text { and } \lim _{x \rightarrow 3} f(x)$$

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 \([1,4]\).

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

Evaluate \(\lim _{x \rightarrow 5} \sqrt{x^{2}-9}\).

Discontinuities from a graph Determine the points at which the following functions \(f\) have discontinuities. At each point of discontinuity, state the conditions in the continuity checklist that are violated.

The position of an object moving vertically along a line is given by the function \(s(t)=-16 t^{2}+128 t .\) Find the average velocity of the object over the following intervals. a. [1,4] b. [1,3] c. [1,2] d. \([1,1+h],\) where \(h>0\) is a real number

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

Suppose $$ f(x)=\left\\{\begin{array}{ll} 4 & \text { if } x \leq 3 \\ x+2 & \text { if } x>3 \end{array}\right. $$ Compute \(\lim _{x \rightarrow 3} f(x)\) and \(\lim _{x \rightarrow 3^{+}} f(x)\).

Average velocity The position of an object moving vertically along a line is given by the function \(s(t)=-4.9 t^{2}+30 t+20\) Find the average velocity of the object over the following intervals. a. [0,3] b. [0,2] c. [0,1] d. \([0, h],\) where \(h>0\) is a real number.

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 \(\lim _{x \rightarrow 2} \frac{x^{2}-4}{x-2}\)