Question

True or False The limit of a function as \(x\) approaches \(c\) depends on the value of the function at \(c\).

Question

True or False In the \(\epsilon\)-\(\delta \) definition of a limit, we require \(0 \lt \vert x-c\vert \) to ensure that \(x\neq c.\)

Question

True or False In an \(\epsilon \)-\(\delta \) proof of a limit, the size of \(\delta \) usually depends on the size of \(\epsilon.\)

Question

True or False When proving \(\lim\limits_{x\rightarrow c}f( x) =L\) using the \(\epsilon \)-\(\delta \) definition of a limit, we try to find a connection between \(\vert f( x) -c\vert \) and \(\vert x-c\vert \).

Question

True or False Given any \(\epsilon \gt0\), suppose there is a \(\delta \gt0\) so that whenever \(0\lt\vert x-c\vert \lt\delta \), then \(\vert f(x)-L\vert \lt\epsilon\). Then \( \lim\limits_{x\rightarrow c}f(x)=L\).

Question

True or False A function \(f\) has a limit \(L\) at infinity, if for any given \(\epsilon \gt0\), there is a positive number \(M\) so that whenever \(x\gt M\), then \(\vert f(x)-L\vert \gt\epsilon\).

Question

\(\lim\limits_{x\rightarrow 1}(2x)=2, \quad \epsilon =0.01\)

Question

\(\lim\limits_{x\rightarrow 2}(-3x)=-6, \quad \epsilon =0.01\)

Question

\(\lim\limits_{x\rightarrow 2}(6x-1)=11\) \(\epsilon =\dfrac{1}{2}\)

Question

\(\lim\limits_{x\rightarrow -3}(2-3x)=11\) \(\epsilon =\dfrac{1}{3}\)

Question

\(\lim\limits_{x\rightarrow 2}\left( -\dfrac{1}{2}x+5\right) =4\) \(\epsilon =0.01\)

Question

\(\lim\limits_{x\rightarrow \frac{5}{6}}\left( 3x+\dfrac{1}{2}\right) =3\) \(\epsilon =0.3\)

Question

For the function \(f(x)=4x-1\), we have \(\lim\limits_{x \rightarrow 3}f(x)=11\). For each \(\epsilon \gt0\), find a \(\delta \gt0\) so that \[ \hbox{whenever }\quad 0 \lt\vert x-3\vert \lt\delta\qquad \hbox{then } \vert(4x-1)-11\vert \lt\epsilon \]

1. \(\epsilon =0.1\)
2. \(\epsilon =0.01\)
3. \(\epsilon =0.001\)
4. \(\epsilon \gt0\) is arbitrary

Question

For the function \(f(x)=2-5x\), we have \(\lim\limits_{x \rightarrow -2}f(x)=12\). For each \(\epsilon \gt0\), find a \(\delta >0\) so that \[ \hbox{whenever } 0 \lt\vert x+2\vert \lt\delta \qquad \hbox{then } \vert (2-5x)-12\vert \lt\epsilon \]

1. \(\epsilon =0.2\)
2. \(\epsilon =0.02\)
3. \(\epsilon =0.002\)
4. \(\epsilon \gt0\) is arbitrary

Question

For the function \(f(x)=\dfrac{x^{2}-9}{x+3}\), we have \( \lim\limits_{x\rightarrow -3} f(x)=-6\). For each \(\epsilon \gt0\), find a \(\delta >0\) so that \[ \hbox{whenever }\quad 0 \lt\vert x+3\vert \lt\delta \qquad \hbox{then } \left\vert \dfrac{x^{2}-9}{x+3}- ( -6) \right\vert \lt\epsilon \]

1. \(\epsilon =0.1\)
2. \(\epsilon =0.01\)
3. \(\epsilon \gt0\) is arbitrary

Question

For the function \(f(x)=\dfrac{x^{2}-4}{x-2}\), we have \(\lim\limits_{x\rightarrow 2}f(x)=4\). For each \(\epsilon \gt0\), find a \(\delta \gt0\) so that \[ \hbox{whenever } 0 \lt\vert x-2\vert \lt\delta \qquad \hbox{then } \left\vert \dfrac{x^{2}-4}{x-2}-4\right\vert \lt\epsilon \]

1. \(\epsilon =0.1\)
2. \(\epsilon =0.01\)
3. \(\epsilon \gt0\) is arbitrary

Question

\(\lim\limits_{x\rightarrow 2}(3x)=6\)

Question

\(\lim\limits_{x\rightarrow 3}(4x)=12\)

Question

\(\lim\limits_{x\rightarrow 0}(2x+5)=5\)

Question

\(\lim\limits_{x\rightarrow -1}(2-3x)=5\)

Question

\(\lim\limits_{x\rightarrow -3}(-5x+2)=17\)

Question

\(\lim\limits_{x\rightarrow 2}(2x-3)=1\)

Question

\(\lim\limits_{x\rightarrow 2}(x^{2}-2x)=0\)

Question

\(\lim\limits_{x\rightarrow 0}(x^{2}+3x)=0\)

Question

\(\lim\limits_{x\rightarrow 1}\dfrac{1+2x}{3-x}=\dfrac{3}{2}\)

Question

\(\lim\limits_{x \rightarrow 2}\dfrac{2x}{4+x}=\dfrac{2}{3}\)

Question

\(\lim\limits_{x\rightarrow 0}\sqrt[3]{{x}}=0\)

Question

\(\lim\limits_{x\rightarrow 1}\sqrt{2-x}=1\)

Question

\(\lim\limits_{x\rightarrow -1}x^{2}=1\)

Question

\(\lim\limits_{x\rightarrow 2}x^{3}=8\)

Question

\(\lim\limits_{x\rightarrow 3}\dfrac{1}{x}=\dfrac{1}{3}\)

Question

\(\lim\limits_{x\rightarrow 2}\dfrac{1}{x^{2}}=\dfrac{1}{4}\)

Question

Use the \(\epsilon\)-\(\delta\) definition of a limit to show that the statement \(\lim\limits_{x\rightarrow 3}(3x-1)=12\) is false.

Question 34

Use the \(\epsilon\)-\(\delta\) definition of a limit to show that the statement \(\lim\limits_{x\rightarrow -2}(4x)=-7\) is false.

Question 35

Show that \(\left\vert \dfrac{{1}}{{x^{2}\,+\,9}}-\dfrac{1}{18} \right\vert \lt\dfrac{{7}}{{234}}\,\vert x-3\vert\) if \(2 \lt x\lt4\). Use this to show that \({\lim\limits_{x\rightarrow 3}}{\dfrac{{1}}{{ x^{2}+9}}}=\dfrac{1}{18}.\)

Question 36

Show that \(\vert (2+x)^{2}-4\vert \leq 5\,\vert x\vert\) if -1 \(\lt x\lt 1\). Use this to show that \(\lim\limits_{x\rightarrow 0}(2+x)^{2}=4\).

Question 37

Show that \({\left\vert {{\dfrac{{1}}{{x^{2}+9}}}-{\dfrac{{1}}{{ 13}}}}\right\vert }\leq {\dfrac{{1}}{{26}}\,}\left\vert \,{x-2}\right\vert\) if \(1 \lt x \lt3\). Use this to show that \({\lim\limits_{x\rightarrow 2}\,}{\dfrac{{1}}{{x^{2}+9}}}={\dfrac{{1}}{{13}}}.\)

Question

Use the \(\epsilon-\delta\) definition of a limit to show that \({\lim\limits_{x\rightarrow 1}}x^{2}\neq 1.31.\) (Hint: Use \(\epsilon = 0.1\).)

Question

If \(m\) and \(b\) are any constants, prove that \[ \lim\limits_{x\rightarrow c}( {mx+b}) =mc+b \]

Question

Verify that if \(x\) is restricted so that \(\vert x-2\vert \lt\dfrac{1}{3}\) in the proof of \(\lim\limits_{x\rightarrow 2}x^{2}=4\), then the choice for \(\delta\) would be less than or equal to the smaller of \(\dfrac{1}{3}\) and \(\dfrac{3\epsilon }{13}\); that is, \(\delta \leq \min \left\{ \dfrac{1}{3},\dfrac{3\epsilon }{13}\right\}\).

Question

For \(x\neq 3\), how close to \(3\) must \(x\) be to guarantee that \(2x-1\) differs from 5 by less than 0.1?

Question

For \(x\neq 0\), how close to \(0\) must \(x\) be to guarantee that \(3^{x}\) differs from \(1\) by less than 0.1?

Question

Prove that if \(\lim\limits_{x\rightarrow c}f(x)\lt0\), then there is an open interval around \(c\) for which \(f(x)\lt0\) everywhere in the interval, except possibly at \(c\).

Question

Use the \(\epsilon-\delta\) definition of a limit at infinity to prove that \[ \lim\limits_{x\rightarrow -\infty }\dfrac{1}{x}=0. \]

Question

Use the \(\epsilon \)-\(\delta \) definition of a limit at infinity to prove that \[ \lim\limits_{x\rightarrow \infty }\left( -\dfrac{1}{\sqrt{x}}\right) =0. \]

Question

For \(\lim\limits_{x\rightarrow -\infty }\dfrac{1}{x^{2}} =0\), find a value of \(N\) that satisfies the \(\epsilon\) - \(\delta\) definition of limits at infinity for \(\epsilon = 0.1\).

Question

Use the \(\epsilon\) - \(\delta\) definition of limit to prove that no number \(L\) exists so that \(\lim\limits_{x\rightarrow 0}\) \(\dfrac{1 }{x}=L\).

Question

Explain why in the \(\epsilon\) - \(\delta\) definition of a limit, the inequality \(0 \lt \vert x-c\vert \lt\delta\) has two strict inequality symbols.

Question

The \(\epsilon\) - \(\delta\) definition of a limit states, in part, that \(f\) is defined everywhere in an open interval containing \(c\), except possibly at \(c\). Discuss the purpose of including the phrase, except possibly at c, and why it is necessary.

Question

In the \(\epsilon\) - \(\delta\) definition of a limit, what does \(\epsilon\) measure? What does \(\delta\) measure? Give an example to support your explanation.

Question

Discuss \(\lim\limits_{x\rightarrow 0}f( x)\) and \(\lim\limits_{x\rightarrow 1}f( x)\) if \begin{equation*} f(x) =\left\{ \begin{array}{l@{\quad}l} x^{2} & \hbox{if \(x\) is rational} \\ 0 & \hbox{if \(x\) is irrational} \end{array}. \right. \end{equation*}

Question

Discuss \(\lim\limits_{x\rightarrow 0}f( x)\) if \begin{equation*} f( x) =\left\{ \begin{array}{c@{\quad}l} x^{2} & \hbox{if \(x\) is rational} \\ \tan x & \hbox{if \(x\) is irrational} \end{array}. \right. \end{equation*}

Question

Use the \(\epsilon\)- \(\delta\) definition of limit to prove that \(\lim\limits_{x\rightarrow 1}(4x^{3}+3x^{2}-24x+22)=5\).

Question

If \(\lim\limits_{x\rightarrow c}f(x)=L\) and \(\lim\limits_{x\rightarrow c}g(x)=M\), prove that \(\lim\limits_{x\rightarrow c}[f(x)+g(x)]=L+M\). Use the \(\epsilon\) - \(\delta\) definition of limit.

Question

For \(\lim\limits_{x\rightarrow \infty }\dfrac{2-x}{\sqrt{ 5+4x^{2}}}=-\dfrac{1}{2}\), find a value of \(M\) that satisfies the \(\epsilon\) - \(\delta\) definition of limits at infinity for \(\epsilon =0.01\).

Question

Use the \(\epsilon\) - \(\delta\) definition of a limit to prove that the linear function \(f(x)=ax+b\) is continuous everywhere.

Question

Show that the function \(f( x) =\left\{ \begin{array}{l@{\quad}ll} 0 & \hbox{if} & x\hbox{ is rational} \\ x & \hbox{if} & x\hbox{ is irrational} \end{array} \right.\) is continuous only at \(x=0\).

Question

Suppose that \(f\) is defined on an interval \((a,b)\) and there is a number \(K\) so that \(\left\vert f(x)-f(c)\right\vert \leq K\vert x-c\vert\) for all \(c\) in \((a,b)\) and \(x\) in \((a,b)\). Such a constant \(K\) is called a Lipschitz constant. Find a Lipschitz constant for \(f(x)=x^{3}\) on \((0,2)\).