Limits

Exercises 3–6, $\underset{x\to c}{\lim }f\left(x\right)=L$In  and $\underset{x\to c}{\lim }g\left(x\right)=M$ for some

real numbers L and M. What, if anything, can you say about $\underset{x\to c}{\lim }\frac{f\left(x\right)}{g\left(x\right)}$ in each case?

In Exercises 3–6, $\underset{x\to c}{\lim }f\left(x\right)=L$ and $\underset{x\to c}{\lim }g\left(x\right)=M$ for some

real numbers L and M. What, if anything, can you say about$\underset{x\to c}{\lim }\frac{f\left(x\right)}{g\left(x\right)}$ in each case?

$L=0$ and $M=0$.

Determine which of the given forms are indeterminate. For each form that is not indeterminate, describe the behavior of a limit of that form. 

$\infty +\infty , \infty -\infty , \infty +1, 0+\infty$.

Determine which of the given forms are indeterminate. For each form that is not indeterminate, describe the behavior of a limit of that form.

$\infty ·\infty 0·\infty 5·\infty 5·0 0.0$

Determine which of the given forms are indeterminate. For each form that is not indeterminate, describe the behavior of a limit of that form.

$\frac{0}{0},\frac{0}{\infty },\frac{\infty }{0},\frac{1}{0},\frac{0}{1},\frac{\infty }{1},\frac{1}{\infty },\frac{\infty }{\infty }$

Determine which of the given forms are indeterminate. For each form that is not indeterminate, describe the behavior of a limit of that form.

$0^1,0^0,0^{\infty },1^{\infty },{\infty }^1,{\infty }^0,{\infty }^{\infty }$

Describe in terms of large and small numbers why it makes intuitive sense that limits of the form $\left(a\right)\frac{1}{\infty } \left(b\right)\frac{0}{\infty } \left(c\right)\frac{0}{1}$must equal 0.

Describe in terms of large and small numbers why it makes intuitive sense that limits of the form (a) $0^{\infty }$ (b) $\left(b\right)0^1$must equal 0.

Describe in terms of large and small numbers why it makes intuitive sense that limits of the form (a) $\frac{1}{0^{+}}$ (b) $\frac{\infty }{0^{+}}$(c) $\frac{\infty }{1}$ must be infinite.

Describe in terms of large and small numbers why it makes intuitive sense that limits of the form $\left(a\right) \infty +\infty \left(b\right) \infty ·\infty \left(c\right) {\infty }^1$ must be infinite.

To prove that the limit forms in Theorem 1.33 are indeterminate, we need only list explicit examples of limits that do and do not exist for each form. Do so for each of the limit forms

from Exercises 15–21. For the last three forms you may want to experiment with a graphing utility to find your examples.

$\frac{0}{0}$ that approaches $\left(a\right) 0 \left(b\right) 2 \left(c\right) \infty$

To prove that the limit forms in Theorem \(1.33\) are indeterminate, we need only list explicit examples of limits that do and do not exist for each form. Do so for each of the limit forms from Exercises \(15–21\). For the last three forms you may want to experiment with a graphing utility to find your examples.

\( \frac{0}{0}\) that approaches (a) \(0\)   (b) \(2\)   (c) \(\infty \)

To prove that the limit forms in Theorem 1.33 are indeterminate, we need only list explicit examples of limits that do and do not exist for each form. Do so for each of the limit forms

from Exercises 15–21. For the last three forms you may want to experiment with a graphing utility to find your examples.

$\frac{\infty }{\infty }$that approaches $\left(a\right)1 \left(b\right) 6 \left(c\right) \infty$

To prove that the limit forms in Theorem 1.33 are indeterminate, we need only list explicit examples of limits that do and do not exist for each form. Do so for each of the limit forms

from Exercises 15–21. For the last three forms you may want to experiment with a graphing utility to find your examples.

$\frac{\infty }{\infty }$that approaches (a) 1 (b) 6 (c)

To prove that the limit forms in Theorem \(1.33\) are indeterminate, we need only list explicit examples of limits that do and do not exist for each form. Do so for each of the limit forms. For the last three forms you may want to experiment with a graphing utility to find your examples. 

\(0\cdot \infty\) that approaches (a) \(0\)   (b) \(1\)    (c) \(\infty\)

$\infty -\infty$that approaches (a) $0$ (b) $5,$ (c) $\infty .$

$0^0$ that approaches (a) $1,$(b) $0,$ (c) $\infty .$

${\infty }^0$ that approaches (a) $1,$ (b) $2,$ (c) $\infty .$

Find the equation of a rational function that could have the graph shown. Take into account roots, holes, and vertical and horizontal asymptotes when constructing your function. 

Find the roots, discontinuous, and horizontal and vertical asymptotes of the function in Exercises 23-24. Support your answers by explicitly computing any relevant limits.

$f\left(x\right)=\frac{x^2-2x-3}{x-3}$

Find the roots, discontinuous, and horizontal and vertical asymptotes of the function in Exercises 23-24. Support your answers by explicitly computing any relevant limits. 

$f\left(x\right)=\frac{2x^2-1}{x^2-2x+1}$

Find the roots, discontinuous, and horizontal and vertical asymptotes of the function in Exercises 23-24. Support your answers by explicitly computing any relevant limits.  

$f\left(x\right)=\frac{\left(x+1\right)\left(x-2\right)}{\left(x-2\right)\left(x+2\right)}$

Find the roots, discontinuous, and horizontal and vertical asymptotes of the function in Exercises 23-24. Support your answers by explicitly computing any relevant limits.   

$f\left(x\right)=\frac{\left(x+1\right){\left(x-2\right)}^2}{\left(x-2\right)\left(x+2\right)}$

Find the roots, discontinuous, and horizontal and vertical asymptotes of the function in Exercises 23-24. Support your answers by explicitly computing any relevant limits.    

$f\left(x\right)=\frac{\left(x+1\right)\left(x-2\right)}{{\left(x-2\right)}^2\left(x+2\right)}$

Find the roots, discontinuities, and horizontal and vertical asymptotes of the functions in Exercises 23–34. Support your answers by explicitly computing any relevant limits.

$f\left(x\right)=\frac{\left(x+1\right)\left(x-2\right)}{\left(x-2\right){\left(x+2\right)}^2}$

Find the roots, discontinuities, and horizontal and vertical asymptotes of the functions in Exercises 23–34. Support your answers by explicitly computing any relevant limits.

$f\left(x\right)=\frac{2}{4+e^{-2x}}$

Find the roots, discontinuities, and horizontal and vertical asymptotes of the functions in Exercises 23–34. Support your answers by explicitly computing any relevant limits.

$f\left(x\right)=\frac{1}{2+3^x}$

Find the roots, discontinuities, and horizontal and vertical asymptotes of the functions in Exercises 23–34. Support your answers by explicitly computing any relevant limits.

$f\left(x\right)=\frac{2^x-4^x}{3^x}$

Find the roots, discontinuities, and horizontal and vertical asymptotes of the functions in Exercises 23–34. Support your answers by explicitly computing any relevant limits.

$f\left(x\right)=\frac{4^x-6\left(2^x\right)+5}{1-2^x}$

Find the roots, discontinuities, and horizontal and vertical asymptotes of the functions in Exercises 23–34. Support your answers by explicitly computing any relevant limits.

$f\left(x\right)={\tan }^{-1}\left(3x\right)+1$

Find the roots, discontinuities, and horizontal and vertical asymptotes of the functions in Exercises 23–34. Support your answers by explicitly computing any relevant limits.

$f\left(x\right)=\frac{1}{{\tan }^{-1}x}$

Calculate each limit in Exercises 35-80.

$\underset{x\to \hspace{0.17em}0}{\lim }\left(-4x^{-3}\right)$

Calculate each limit in Exercises 35-80.

$\underset{x\to \hspace{0.17em}0}{\lim }\left(2x^{-\frac{3}{4}}\right)$

Calculate each limit in Exercises 35-80.

$\underset{x\to \infty }{\lim }\left(2x^{-\frac{4}{3}}\right)$

Calculate each limit in Exercises 35-80.

$\underset{x\to \infty }{\lim }\left(-5x^{\frac{3}{5}}\right)$

Calculate each limit in Exercises 35-80.

$\underset{x\to \infty }{\lim }\left(\sqrt{x}-x\right)$

Calculate each limit in Exercises 35-80.

$\underset{x\to \infty }{\lim }\left(x^4-x^5\right)$

Calculate each limit in Exercises 35-80.

$\underset{x\to \infty }{\lim }\left(-3x^5+4x+11\right)$

Calculate each limit in Exercises 35-80.

$\underset{x\to -\infty }{\lim }\left(5-2x+3x^3\right)$

Calculate each limit in Exercises 35–80. 

$\underset{x\to -4}{lim}\frac{x^2+8x+16}{{(x+4)}^2(x+1)}$

Calculate each limit in Exercises 35–80.  

$\underset{x\to 2}{lim}\frac{x+1}{{(x-2)}^2}$

Calculate each limit in Exercises 35–80. 

$\underset{x\to -4}{\lim }\frac{x+4}{x^2+8x+16}$

Calculate each limit in Exercises 35–80.

$\underset{x\to 0}{\lim } \frac{x}{x^2-x}$

Calculate each limit in Exercises 35–80.  

$\underset{x\to \infty }{\lim }\frac{{(3x+1)}^2(x-1)}{1-x^3}$

Calculate each limit in Exercises 35–80. 

$\underset{x\to -\infty }{\lim }\frac{1-2x^2}{(3-x)(3+4x)}$

Calculate each limit in Exercises

$\underset{x\to 0^{+}}{\lim }\left(x^{-\frac{1}{3}}-x^{-\frac{1}{2}}\right)$

Calculate each limit in Exercises 35–80.  

$\underset{x\to \infty }{\lim }(x^{-\frac{1}{3}}-x^{-\frac{1}{2}})$

Calculate each limit in Exercises 35–80.
$\underset{x\to \infty }{lim}\frac{x^{-3}}{x^2-x^{-1}}$

Calculate each limit in Exercises 35–80.
$\underset{x\to 0^{+}}{lim}\frac{x^{-3}}{x^2-x^{-1}}$

Calculate each limit in Exercises 35–80.
$\underset{x\to 0^{+}}{lim}\frac{x^{{\displaystyle \raisebox{1ex}{$7$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}-x^{{\displaystyle \raisebox{1ex}{$8$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}}{x^2}$