Find the limit (if it exists). $$\lim _ { x \rightarrow 5 } \frac { \sqrt { x + 4 } - 3 } { x - 5 }$$
A) 6
B) 1
C) 0
D) $\frac { 1 } { 6 }$
E) Limit does not exist.

Question

Quizplus · Accepted Answer

To evaluate the limit, we can use the conjugate and simplify:
\begin{align*}
\lim_{x	o 5} \frac{\sqrt{x+4}-3}{x-5} &= \lim_{x	o 5} \frac{(\sqrt{x+4}-3)(\sqrt{x+4}+3)}{(x-5)(\sqrt{x+4}+3)} \
&= \lim_{x	o 5} \frac{(x+4)-9}{(x-5)(\sqrt{x+4}+3)} \
&= \lim_{x	o 5} \frac{x-5}{(x-5)(\sqrt{x+4}+3)} \
&= \lim_{x	o 5} \frac{1}{\sqrt{x+4}+3} \
&= \frac{1}{\sqrt{9}+3} \
&= \boxed{\frac{1}{6}}
\end{align*}

Find the Limit (If It Exists) lim⁡x→5x+4−3x−5\lim _ { x \rightarrow 5 } \frac { \sqrt { x + 4 } - 3 } { x - 5 }x→5lim​x−5x+4​−3​

Find the Limit (If It Exists) $\lim _ { x \rightarrow 5 } \frac { \sqrt { x + 4 } - 3 } { x - 5 }$