Single Variable Calculus, Chapter 2, Review Exercises, Section Review Exercises, Problem 9

Determine $\displaystyle \lim \limits_{r \to 9} \frac{\sqrt{r}}{(r - 9)^4}$


$
\begin{equation}
\begin{aligned}



& \lim \limits_{r \to 9} \frac{\sqrt{r}}{(r - 9)^4} = \lim \limits_{r \to 9} \frac{\sqrt{9}}{(9 - 9)^4} = \frac{3}{0}
&& \text{By substituting value of $r$ to the equation we will get $\displaystyle \frac{3}{0}$. So the limit is undefined.}

\end{aligned}
\end{equation}
$


So we set values for $r$ as $r$ approaches 9 from both sides.


$\begin{array}{|c|c|}
\hline\\
r & f(r) \\
\hline\\
8.99 & 299833287 \\
8.999 & 2.999833329 \, x \, 10^{12} \\
9.001 & 3.000166662 \, x \, 10^{12} \\
9.01 & 300166620\\
\hline
\end{array} $

Referring to the table, the values of $f(r)$ became more positively large number as $r$ approaches 9 from both sides.

Therefore,

$\fbox{$ \lim \limits_{r \to 9} \displaystyle \frac{\sqrt{r}}{(r - 9)^4 } = \infty $}$