1. Limits and Continuity

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 _____ .

Where is the function continuous? Differentiable? Use the graph of f in the figure to do the following. <IMAGE>

a. Find the values of x in (0, 3) at which f is not continuous.

Use the graph of f in the figure to do the following. <IMAGE>

a. Find the values of x in (-2,2) at which f is not continuous.

Continuity of a piecewise function Let g(x) = <matrix 2x1> For what values of a is g continuous?

Continuous Extension

Explain why the function ƒ(𝓍) = sin(1/𝓍) has no continuous extension to 𝓍 = 0.