1. Limits and Continuity

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 $1$. 

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 $1$.

Find the intervals on which the following functions are continuous. Specify right- or left-continuity at the finite endpoints.

$h\left(x\right)=\frac{2x}{x^3-25x}$

Find the intervals on which the following functions are continuous. Specify right- or left-continuity at the finite endpoints.

$g\left(x\right)=\cos \left(e^x\right)$

Suppose the rental cost for a snowboard is $25 for the first day (or any part of the first day) plus $15 for each additional day (or any part of a day).

e. For what values of t is f continuous? Explain.

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

Determine values of the constants $a$ and $b$ , if possible, for which $g$ is continuous at $x=1$.

Complete the following sentences in terms of a limit.

a. A function is continuous from the left at a if _____.

Complete the following sentences in terms of a limit.

b. A function is continuous from the right at a if _____ .