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

Evaluate each limit. 

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

Evaluate each limit. 

$\underset{x\to 3}{\lim }\sqrt{x^2+7}$

Evaluate each limit. 

$\underset{x\to \frac{\pi }{2}}{\lim }\frac{\sin \left(x\right)-1}{\sqrt{\sin \left(x\right)}-1}$

Assume you invest $250 at the end of each year for 10 years at an annual interest rate of $r$. The amount of money in your account after 10 years is given by $A\left(r\right)=\frac{250({(1+r)}^{10}-1)}{r}$. Assume your goal is to have $3500 in your account after 10 years.

b. Use a calculator to estimate the interest rate required to reach your financial goal.

Find an interval containing a solution to the equation $2x=\cos \left(x\right)$. Use a graphing utility to approximate the solution.

Complete the following steps for each function.

c. State the interval(s) of continuity.

f(x)={2x if x<1

x^2+3x if x≥1; a=1

Let $g\left(x\right)=\begin{cases}x^2+x & \text{if }x<1\\ a & \text{if }x=1\\ 3x+5 & \text{if }x>1\end{cases}$

a. Determine the value of a for which $g$ is continuous from the left at $1$.

Determine whether the following functions are continuous at a. Use the continuity checklist to justify your answer. 

f(x)=2x^2+3x+1 / x^2+5x; a=−5

Determine whether the following functions are continuous at a. Use the continuity checklist to justify your answer. 

f(x)= √x−2; a=1

Determine the interval(s) on which the following functions are continuous. 

p(x)=4x^5−3x^2+1

Determine the interval(s) on which the following functions are continuous. 

p(x)=3x^2−6x+7 / x^2+x+1

Determine the interval(s) on which the following functions are continuous. 

f(x)=x^5+6x+17 / x^2−9

Determine the interval(s) on which the following functions are continuous. 

f(x)=1 / x^2−4

Determine the interval(s) on which the following functions are continuous. 

f(t)=t+2 / t^2−4

Let $g\left(x\right)=\begin{cases}x^2+x & \text{if }x<1\\ a & \text{if }x=1\\ 3x+5 & \text{if }x>1\end{cases}$

b. Determine the value of $a$ for which $g$ is continuous from the right at $1$. 

Evaluate each limit and justify your answer. 

lim x→4 √x^3−2x^2−8x / x−4

Evaluate each limit and justify your answer. 

lim x→1 (x+5x / x+2)^4

Evaluate each limit and justify your answer. 

lim x→∞(2x+1x / x)^3

a. Analyze ${\displaystyle\lim_{x\to\infty}{f(x)}}$ and${\displaystyle\lim_{x\to-\infty}{f(x)}}$ for each function.

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

Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.

lim x→3 x − 3 /|x − 3|

Suppose f(x) lies in the interval (2, 6). What is the smallest value of ε such that |f (x)−4|<ε?

Use the precise definition of a limit to prove the following limits. Specify a relationship between ε and δ that guarantees the limit exists.

lim x→1 x^4=1

The following table gives the position $s\left(t\right)$ of an object moving along a line at time $t$. Determine the average velocities over the time intervals $\left\lbrack1,1.01\right\rbrack$, $\left\lbrack1,1.001\right\rbrack$, and $\left\lbrack1,1.0001^{}\right.]$. Then make a conjecture about the value of the instantaneous velocity at $t=1$. <IMAGE>

Suppose x lies in the interval (1, 3) with x≠2. Find the smallest positive value of δ such that the inequality 0<|x−2|<δ is true.

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)

Suppose |f(x) − 5|<0.1 whenever 0<x<5. Find all values of δ>0 such that |f(x) − 5|<0.1 whenever 0<|x−2|<δ.

Use the precise definition of a limit to prove the following limits. Specify a relationship between ε and δ that guarantees the limit exists.

lim x→1 (8x+5)=13

Use the precise definition of a limit to prove the following limits. Specify a relationship between ε and δ that guarantees the limit exists.

lim x→4 x^2−16 / x−4=8 (Hint: Factor and simplify.)

Use the precise definition of a limit to prove the following limits. Specify a relationship between ε and δ that guarantees the limit exists.

lim x→7 f(x)=9, where f(x)={3x−12 if x≤7

x+2 if x>7