Chapter 1

T/F: If \(\lim _{x \rightarrow 5} f(x)=\infty,\) then we are implicitly stating that the limit exists.

In your own words, describe what it means for a function to be continuous.

What are the three ways in which a limit may fail to exist?

Explain in your own words, without using \(\varepsilon-\delta\) formality, why \(\lim _{x \rightarrow c} b=b\)

What is wrong with the following "definition" of a limit? "The limit of \(f(x)\), as \(x\) approaches \(a\), is \(K^{\prime \prime}\) means that given any \(\delta>0\) there exists \(\varepsilon>0\) such that whenever \(|f(x)-K|<\varepsilon,\) we have \(|x-a|<\delta\)

In your own words, what does it mean to "find the limit of \(f(x)\) as \(x\) approaches \(3^{\prime \prime} ?\)

T/F: If \(\lim _{x \rightarrow \infty} f(x)=5,\) then we are implicitly stating that the limit exists.

In your own words, describe what the Intermediate Value Theorem states.

Explain in your own words, without using \(\varepsilon-\delta\) formality, why \(\lim _{x \rightarrow c} b=b\)

An expression of the form \(\frac{0}{0}\) is called ______.

\(\mathrm{T} / \mathrm{F}:\) If \(\lim _{x \rightarrow 1^{-}} f(x)=-\infty,\) then \(\lim _{x \rightarrow 1^{+}} f(x)=\infty\)

What is a "root" of a function?

T/F: If \(\lim _{x \rightarrow 1^{-}} f(x)=5,\) then \(\lim _{x \rightarrow 1^{+}} f(x)=5\)

T/F: If \(\lim _{x \rightarrow 5} f(x)=\infty\), then \(f\) has a vertical asymptote at \(x=5\)

Given functions \(f\) and \(g\) on an interval \(I\), how can the Bisection Method be used to find a value \(c\) where \(f(c)=g(c)\) ?

T/F: If \(\lim _{x \rightarrow 1} f(x)=5,\) then \(\lim _{x \rightarrow 1^{-}} f(x)=5\)

Describe three situations where \(\lim f(x)\) does not exist.

\(\mathrm{T} / \mathrm{F}: \infty / 0\) is not an indeterminate form.

\(\mathrm{T} / \mathrm{F}:\) If \(f\) is defined on an open interval containing \(c\), and \(\lim _{x \rightarrow c} f(x)\) exists, then \(f\) is continuous at \(c\).

You are given the following information: (a) \(\lim _{x \rightarrow 1} f(x)=0\) (b) \(\lim _{x \rightarrow 1} g(x)=0\) (c) \(\lim _{x \rightarrow 1} f(x) / g(x)=2\) What can be said about the relative sizes of \(f(x)\) and \(g(x)\) as \(x\) approaches \(1 ?\)

Prove the given limit using an \(\varepsilon-\delta\) proof. $$ \lim _{x \rightarrow 4}(2 x+5)=13 $$

In your own words, what is a difference quotient?

List 5 indeterminate forms.

\(\mathrm{T} / \mathrm{F}\) : If \(f\) is continuous at \(c,\) then \(\lim _{x \rightarrow c} f(x)\) exists.

\(\mathrm{T} / \mathrm{F}: \lim _{x \rightarrow 1} \ln x=0 .\) Use a theorem to defend your answer.

Prove the given limit using an \(\varepsilon-\delta\) proof. $$ \lim _{x \rightarrow 5}(3-x)=-2 $$

When \(x\) is near \(0, \frac{\sin x}{x}\) is near what value?

Construct a function with a vertical asymptote at \(x=5\) and a horizontal asymptote at \(y=5\).

\(\mathrm{T} / \mathrm{F}:\) If \(f\) is continuous at \(c,\) then \(\lim _{x \rightarrow c^{+}} f(x)=f(c) .\)

Use the following information to evaluate the given limit, when possible. If it is not possible to determine the limit, state why not. $$ \begin{array}{ll} *\lim _{x \rightarrow 9} f(x)=6, & \lim _{x \rightarrow 6} f(x)=9, \quad f(9)=6 \\ *\lim _{x \rightarrow 9} g(x)=3, & \lim _{x \rightarrow 6} g(x)=3, \quad g(6)=9 \end{array} $$ $$ \lim _{x \rightarrow 9}(f(x)+g(x)) $$

Prove the given limit using an \(\varepsilon-\delta\) proof. $$ \lim _{x \rightarrow 3}\left(x^{2}-3\right)=6 $$

Approximate the given limits both numerically and graphically. $$ \lim _{x \rightarrow 1} x^{2}+3 x-5 $$

Let \(\lim _{x \rightarrow 7} f(x)=\infty\). Explain how we know that \(f\) is/is not continuous at \(x=7\).

Use the following information to evaluate the given limit, when possible. If it is not possible to determine the limit, state why not. $$ \begin{array}{ll} *\lim _{x \rightarrow 9} f(x)=6, & \lim _{x \rightarrow 6} f(x)=9, \quad f(9)=6 \\ *\lim _{x \rightarrow 9} g(x)=3, & \lim _{x \rightarrow 6} g(x)=3, \quad g(6)=9 \end{array} $$ $$ \lim _{x \rightarrow 9}(3 f(x) / g(x)) $$

Prove the given limit using an \(\varepsilon-\delta\) proof. $$ \lim _{x \rightarrow 4}\left(x^{2}+x-5\right)=15 $$

Approximate the given limits both numerically and graphically. $$ \lim _{x \rightarrow 0} x^{3}-3 x^{2}+x-5 $$

Use the following information to evaluate the given limit, when possible. If it is not possible to determine the limit, state why not. $$ \begin{array}{ll} *\lim _{x \rightarrow 9} f(x)=6, & \lim _{x \rightarrow 6} f(x)=9, \quad f(9)=6 \\ *\lim _{x \rightarrow 9} g(x)=3, & \lim _{x \rightarrow 6} g(x)=3, \quad g(6)=9 \end{array} $$ $$ \lim _{x \rightarrow 9}\left(\frac{f(x)-2 g(x)}{g(x)}\right) $$

Prove the given limit using an \(\varepsilon-\delta\) proof. $$ \lim _{x \rightarrow 1}\left(2 x^{2}+3 x+1\right)=6 $$

Approximate the given limits both numerically and graphically. $$ \lim _{x \rightarrow 0} \frac{x+1}{x^{2}+3 x} $$

\(\mathrm{T} / \mathrm{F}:\) The sum of continuous functions is also continuous.

Use the following information to evaluate the given limit, when possible. If it is not possible to determine the limit, state why not. $$ \begin{array}{ll} *\lim _{x \rightarrow 9} f(x)=6, & \lim _{x \rightarrow 6} f(x)=9, \quad f(9)=6 \\ *\lim _{x \rightarrow 9} g(x)=3, & \lim _{x \rightarrow 6} g(x)=3, \quad g(6)=9 \end{array} $$ $$ \lim _{x \rightarrow 6}\left(\frac{f(x)}{3-g(x)}\right) $$

Prove the given limit using an \(\varepsilon-\delta\) proof. $$ \lim _{x \rightarrow 2}\left(x^{3}-1\right)=7 $$

Approximate the given limits both numerically and graphically. $$ \lim _{x \rightarrow 3} \frac{x^{2}-2 x-3}{x^{2}-4 x+3} $$

Prove the given limit using an \(\varepsilon-\delta\) proof. $$ \lim _{x \rightarrow 2} 5=5 $$

Evaluate the given limits using the graph of the function. \(f(x)=x^{2} \sin (\pi x)\) (a) \(\lim _{x \rightarrow-\infty} f(x)\) (b) \(\lim _{x \rightarrow \infty} f(x)\)

Use the following information to evaluate the given limit, when possible. If it is not possible to determine the limit, state why not. $$ \begin{array}{ll} *\lim _{x \rightarrow 9} f(x)=6, & \lim _{x \rightarrow 6} f(x)=9, \quad f(9)=6 \\ *\lim _{x \rightarrow 9} g(x)=3, & \lim _{x \rightarrow 6} g(x)=3, \quad g(6)=9 \end{array} $$ $$ \lim _{x \rightarrow 6} f(g(x)) $$

Prove the given limit using an \(\varepsilon-\delta\) proof. $$ \lim _{x \rightarrow 0}\left(e^{2 x}-1\right)=0 $$

Approximate the given limits both numerically and graphically. $$ \lim _{x \rightarrow 2} \frac{x^{2}+7 x+10}{x^{2}-4 x+4} $$

Evaluate the given limits using the graph of the function. $$ \begin{array}{l} f(x)=\cos (x) \\ \text { (a) } \lim _{x \rightarrow-\infty} f(x) \\ \text { (b) } \lim _{x \rightarrow \infty} f(x) \end{array} $$

Evaluate the given limits of the piecewise defined functions \(f\). \(f(x)=\left\\{\begin{array}{cl}x+1 & x \leq 1 \\ x^{2}-5 & x>1\end{array}\right.\) (a) \(\lim _{x \rightarrow 1^{-}} f(x)\) (c) \(\lim _{x \rightarrow 1} f(x)\) (b) \(\lim _{x \rightarrow 1^{+}} f(x)\) (d) \(f(1)\)