Refer to the graph of the function $f(x)$ to find the given limit if exists. If the limit does not exist, write "DNE."

$\underset{x\to 5}{\lim }f\left(x\right)$

Evaluate the following limits and identify the horizontal asymptotes (if any) for the function $f\left(x\right)=\frac{5x}{25x+3}$:

${\lim }_{x\to \infty }f\left(x\right)$

${\lim }_{x\to -\infty }f\left(x\right)$

Select the correct relationship between $ϵ$ and $\delta$ to prove $\underset{x\to -8}{\lim }\mid 5x\mid =40$ using the $\epsilon -\delta$ definition of a limit.

The radius of a right cylinder having a height of $15\text{ cm}$ and a surface area of $U{\text{ cm}}^2$ is given as $r\left(U\right)=\frac{1}{5}(\sqrt{225+\frac{5U}{\pi }}-15)$. Calculate $\underset{U\to 0^{+}}{\lim }r\left(U\right)$ and provide an interpretation.

Use the following theorem to evaluate $\underset{x\to 0}{\lim }\frac{\sin 39x}{9x}$:

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

On the interval $(0,15),$ locate the points where the function $f$ has discontinuities. For each discontinuity, indicate which continuity conditions are not met.

Let

For what value of $a$ is $g\left(x\right)$ continuous at $x=1$?