Answer:
c. [tex]\frac{1}{2 \sqrt{7} }[/tex]
Step-by-step explanation:
When plugging in zero into the given equation:
[tex]\lim_{x \rightarrow 0} \frac{\sqrt{x + 7} - \sqrt{7} }{x} = \frac{0}{0}[/tex]
Answer is in indeterminate form = use L'Hospital's Rule:
(Derivative of the top / Derivative of the bottom)
[tex]\lim_{x \rightarrow 0} \frac{ \frac{1}{2} (x + 7)^{ \frac{-1}{2}} - 0 }{1}[/tex]
Rearranged equation:
[tex]\lim_{x \rightarrow 0} \frac{1}{2 \sqrt{x + 7} }[/tex]
Plug zero back into equation:
[tex]\lim_{x \rightarrow 0} \frac{1}{2 \sqrt{x + 7} } = \frac{1}{2 \sqrt{0 + 7} } = \frac{1}{2 \sqrt{7} }[/tex]
Answer:
[tex]\lim_{x \rightarrow 0} \frac{\sqrt{x + 7} - \sqrt{7} }{x} = \frac{1}{2 \sqrt{7} }[/tex]
