Question

Find the limit, if it exists (picture below)

Answer 1

Answer:

c. $\frac{1}{2 \sqrt{7} }$

Step-by-step explanation:

When plugging in zero into the given equation:

$\lim_{x \rightarrow 0} \frac{\sqrt{x + 7} - \sqrt{7} }{x} = \frac{0}{0}$

Answer is in indeterminate form = use L'Hospital's Rule:

(Derivative of the top / Derivative of the bottom)

$\lim_{x \rightarrow 0} \frac{ \frac{1}{2} (x + 7)^{ \frac{-1}{2}} - 0 }{1}$

Rearranged equation:

$\lim_{x \rightarrow 0} \frac{1}{2 \sqrt{x + 7} }$

Plug zero back into equation:

$\lim_{x \rightarrow 0} \frac{1}{2 \sqrt{x + 7} } = \frac{1}{2 \sqrt{0 + 7} } = \frac{1}{2 \sqrt{7} }$

Answer:

$\lim_{x \rightarrow 0} \frac{\sqrt{x + 7} - \sqrt{7} }{x} = \frac{1}{2 \sqrt{7} }$