Limits

Limits of basic functions: Fill in the blanks to complete the limit rules that follow. You may assume that k is positive.

$\underset{x\to 0}{\lim }\frac{e^x-1}{x}=\_\_\_\_\_\_\_$

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

Calculate the limit.

Limits of basic functions: Fill in the blanks to complete the limit rules that follow. You may assume that k is positive. 

$\underset{x\to 0}{\lim }{\left(1+x\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$x$}\right.}=\_\_\_\_\_\_$

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

Calculate the limit.

Limits of basic functions: Fill in the blanks to complete the limit rules that follow. You may assume that k is positive. 

$\underset{x\to 0}{\lim }\frac{\sin x}{x}= \_\_\_\_\_\_\_$

Evaluate the limit of the expression

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

Limits of basic functions: Fill in the blanks to complete the limit rules that follow. You may assume that k is positive. 

$\underset{x\to 0}{\lim }\frac{1-\cos x}{x}=\_\_\_\_\_$

$\underset{x\to \infty }{\lim }\frac{\sqrt{x}}{1-\sqrt{x}}$

Calculate the limit.

Limits of combinations: Fill in the blanks to complete the limit

rules that follow. You may assume that k and c are any real

numbers and that both $\underset{x\to c}{\lim }f\left(x\right) \mathrm{and} \underset{\mathrm{x}\to \mathrm{c}}{\lim }g\left(\mathrm{x}\right)$exist.

$\underset{x\to c}{\lim k}f\left(x\right) =\_\_\_\_\_\_\_$

Limits of combinations: Fill in the blanks to complete the limit

rules that follow. You may assume that k and c are any real

numbers and that both $\underset{x\to c}{\lim }f\left(x\right) \mathrm{and} \underset{\mathrm{x}\to \mathrm{c}}{\lim }g\left(\mathrm{x}\right)$ exist.

$\underset{x\to c}{\lim }\left(f\left(x\right)+g\left(x\right)\right)= \_\_\_\_\_\_$

Find the limit by hand.

$\underset{x\to -\infty }{\lim }{e}^x {\tan }^{-1}x$

Limits of combinations: Fill in the blanks to complete the limit

rules that follow. You may assume that k and c are any real

numbers and that both $\underset{x\to c}{\lim }f\left(x\right) \mathrm{and} \underset{\mathrm{x}\to \mathrm{c}}{\lim }g\left(\mathrm{x}\right)$ exist.

$\underset{x\to c}{\lim }\left(f\left(x\right)-g\left(x\right)\right)=\_\_\_\_\_\_$

Find the limit by hand.

$\underset{x\to 0}{\lim }\frac{3x}{\sin 2x}$

Limits of combinations: Fill in the blanks to complete the limit

rules that follow. You may assume that k and c are any real

numbers and that both $\underset{x\to c}{\lim }f\left(x\right) \mathrm{and} \underset{\mathrm{x}\to \mathrm{c}}{\lim }g\left(\mathrm{x}\right)$ exist.

$\underset{x\to c}{\lim }\left(f\left(x\right)g\left(x\right)\right)=\_\_\_\_\_\_$

Find the limit by hand.

$\underset{x\to 0}{\lim }\frac{{\sin }^2 3x}{x}$

Limits of combinations: Fill in the blanks to complete the limit

rules that follow. You may assume that k and c are any real

numbers and that both $\underset{x\to c}{\lim }f\left(x\right) \mathrm{and} \underset{\mathrm{x}\to \mathrm{c}}{\lim }g\left(\mathrm{x}\right)$ exist.

$\underset{x\to c}{\lim }\frac{f\left(x\right)}{g\left(x\right)}=\_\_\_\_\_\_, \mathrm{provided} \mathrm{that} \mathrm{\_\_\_\_\_}$

Find the limit by hand.

$\underset{x\to 0}{\lim }\frac{1-\cos x}{\sin x}$

Limits of combinations: Fill in the blanks to complete the limit

rules that follow. You may assume that k and c are any real

numbers and that both $\underset{x\to c}{\lim }f\left(x\right) \mathrm{and} \underset{\mathrm{x}\to \mathrm{c}}{\lim }g\left(\mathrm{x}\right)$ exist.

$\underset{x\to c}{\lim }f\left(g\right(x\left)\right)=\_\_\_\_\_\_, \mathrm{provided} \mathrm{that} \mathrm{\_\_\_\_\_}$

Find the limit by hand.

$\underset{x\to \infty }{\lim }\sin \left({\tan }^{-1}x\right)$

Indeterminate forms: Identify which of the limit forms listed here are indeterminate. For each form that is not indeterminate, describe the behavior of a limit of that form.

$\frac{1}{0^{+}}$

Find the limit by hand.

$\underset{x\to \infty }{\lim }{\left(1+\frac{1}{x}\right)}^x$

Indeterminate forms: Identify which of the limit forms listed here are indeterminate. For each form that is not indeterminate, describe the behavior of a limit of that form. 

$\frac{0}{1}$

Find the limit by hand.

$\underset{x\to 0}{\lim } \frac{x^2}{e^x-1}$

Indeterminate forms: Identify which of the limit forms listed here are indeterminate. For each form that is not indeterminate, describe the behavior of a limit of that form. 

$\frac{\infty }{\infty }$

Find the limit by hand.

$\underset{x\to \infty }{\lim } \sin x$

Indeterminate forms: Identify which of the limit forms listed here are indeterminate. For each form that is not indeterminate, describe the behavior of a limit of that form.

$\infty \left(-\infty \right)$

Find the limit by hand.

Indeterminate forms: Identify which of the limit forms listed here are indeterminate. For each form that is not indeterminate, describe the behavior of a limit of that form. 

$0^1$

Find the limit by hand.

$\underset{x\to \infty }{\lim } \frac{1}{x} \sin x$

Indeterminate forms: Identify which of the limit forms listed here are indeterminate. For each form that is not indeterminate, describe the behavior of a limit of that form. 

${\infty }^{\infty }$

Find the limit by hand.

$\underset{x\to 0}{\lim } x \sin \frac{1}{x}$

Indeterminate forms: Identify which of the limit forms listed here are indeterminate. For each form that is not indeterminate, describe the behavior of a limit of that form.  

$\frac{1}{0^{-}}$

Indeterminate forms: Identify which of the limit forms listed here are indeterminate. For each form that is not indeterminate, describe the behavior of a limit of that form. 

$\frac{0}{0}$

Indeterminate forms: Identify which of the limit forms listed here are indeterminate. For each form that is not indeterminate, describe the behavior of a limit of that form. 

$\frac{\infty }{0^{+}}$

Indeterminate forms: Identify which of the limit forms listed here are indeterminate. For each form that is not indeterminate, describe the behavior of a limit of that form. 

$\infty +\infty$

Indeterminate forms: Identify which of the limit forms listed here are indeterminate. For each form that is not indeterminate, describe the behavior of a limit of that form.

$0^{\infty }$.

Indeterminate forms: Identify which of the limit forms listed here are indeterminate. For each form that is not indeterminate, describe the behavior of a limit of that form.

${\infty }^1.$

Indeterminate forms: Identify which of the limit forms listed here are indeterminate. For each form that is not indeterminate, describe the behavior of a limit of that form.

$\frac{1}{\infty }.$

Indeterminate forms: Identify which of the limit forms listed here are indeterminate. For each form that is not indeterminate, describe the behavior of a limit of that form.

$\frac{\infty }{1}.$

Indeterminate forms: Identify which of the limit forms listed here are indeterminate. For each form that is not indeterminate, describe the behavior of a limit of that form.

$0\times \infty$.

Indeterminate forms: Identify which of the limit forms listed here are indeterminate. For each form that is not indeterminate, describe the behavior of a limit of that form. 

$\infty -\infty$.

Indeterminate forms: Identify which of the limit forms listed here are indeterminate. For each form that is not indeterminate, describe the behavior of a limit of that form. 

$0^{-\infty }$.

Indeterminate forms: Identify which of the limit forms listed here are indeterminate. For each form that is not indeterminate, describe the behavior of a limit of that form. 

$\frac{1}{-\infty }$.

Indeterminate forms: Identify which of the limit forms listed here are indeterminate. For each form that is not indeterminate, describe the behavior of a limit of that form.  

$\frac{0}{\infty }.$

Indeterminate forms: Identify which of the limit forms listed here are indeterminate. For each form that is not indeterminate, describe the behavior of a limit of that form.  

$\infty \times \infty .$

Indeterminate forms: Identify which of the limit forms listed here are indeterminate. For each form that is not indeterminate, describe the behavior of a limit of that form.

$0^0.$

Indeterminate forms: Identify which of the limit forms listed here are indeterminate. For each form that is not indeterminate, describe the behavior of a limit of that form.

${\infty }^0.$

Indeterminate forms: Identify which of the limit forms listed here are indeterminate. For each form that is not indeterminate, describe the behavior of a limit of that form.

${\infty }^{-\infty }.$

Indeterminate forms: Identify which of the limit forms listed here are indeterminate. For each form that is not indeterminate, describe the behavior of a limit of that form. 

$1^{\infty }$.

For each given function f, find a real number a that makes f continuous at $x=0,$if possible.

$$\begin{gathered} \left(a\right) f\left(x\right)=\left\{\begin{array}{ll}3x+1,& \mathrm{if} x<0\\ 2x+a,& \mathrm{if} x\ge 0\end{array}\right. \\ \left(b\right) f\left(x\right)=\left\{\begin{array}{ll}\frac{a}{x+2},& \mathrm{if} x<0\\ 3,& \mathrm{if} x=0\\ ax+1,& \mathrm{if} x>0\end{array}\right. \end{gathered}$$