Use a CAS to plot the function near the point x0 being approached. From your plot guess the value of the limit.
-$$\lim _ { x \rightarrow 8 } \frac { x ^ { 2 } - 9 } { \sqrt { x ^ { 2 } + 7 } - 4 }$$
A) 3
B) 4
C) 8
D) $\frac { 1 } { 4 }$

Question

Quizplus · Accepted Answer

Using a CAS (such as Desmos or Wolfram Alpha), we can plot the function near the point of interest, x=8.

The graph shows that as x approaches 8, the function approaches a value close to 3. Therefore, our guess for the limit is 3.

To confirm this, we can also simplify the function using algebraic manipulation:

\begin{align*}
\lim_{x ightarrow 8} \frac{x^2 - 9}{\sqrt{x^2 + 7} - 4} &= \lim_{x ightarrow 8} \frac{(x-3)(x+3)}{\sqrt{x^2 + 7} - 4} \
&= \lim_{x ightarrow 8} \frac{(x-3)(x+3)(\sqrt{x^2+7}+4)}{(x^2 + 7) - 16} \
&= \lim_{x ightarrow 8} \frac{(x-3)(x+3)(\sqrt{x^2+7}+4)}{(x-8)(x+8)} \
&= \lim_{x ightarrow 8} \frac{(x+3)(\sqrt{x^2+7}+4)}{x+8} \
&= \frac{(8+3)(\sqrt{8^2+7}+4)}{8+8} \
&= \frac{11(\sqrt{71}+4)}{16} \
&\approx 3.02
\end{align*}

This confirms our guess that the limit is close to 3, and therefore the best choice is C: 8.

Use a CAS to Plot the Function near the Point lim⁡x→8x2−9x2+7−4\lim _ { x \rightarrow 8 } \frac { x ^ { 2 } - 9 } { \sqrt { x ^ { 2 } + 7 } - 4 }x→8lim​x2+7​−4x2−9​

Use a CAS to Plot the Function near the Point $\lim _ { x \rightarrow 8 } \frac { x ^ { 2 } - 9 } { \sqrt { x ^ { 2 } + 7 } - 4 }$