$\text { Evaluate the limit } \lim _ { x \rightarrow 0 ^ { + } } \left( e ^ { x } + 5 x \right) ^ { 5 / x } \text { using L'Hopital's Rule if necessary. }$
A) $e ^ { 5 }$
B) $e ^ { 10 }$
C) 1
D) $e ^ { 30 }$
E) $e + 5$

Question

$\text { Evaluate the limit } \lim _ { x \rightarrow 0 ^ { + } } \left( e ^ { x } + 5 x \right) ^ { 5 / x } \text { using L'Hopital's Rule if necessary. }$ 
A) $e ^ { 5 }$
B) $e ^ { 10 }$
C) 1
D) $e ^ { 30 }$
E) $e + 5$

Quizplus · Accepted Answer

Using the limit formula $e^{\lim_{x 	o a}f(x)}=\lim_{x	o a}e^{f(x)}$ when the original limit is of the form $\lim_{x	o a}f(x)^{g(x)}$, we can rewrite the given limit as $$e^{\lim_{x 	o 0^+}\frac{5}{x}\ln(e^x+5x)}.$$Then, applying L'Hopital's Rule gives us $$\lim_{x 	o 0^+}\frac{5(e^x+5x)(e^x)+5}{e^x+5x}$$which simplifies to 30. Therefore, the limit is $e^{30}$.

A) e5e ^ { 5 }e5 B) e10e ^ { 10 }e10

A) $e ^ { 5 }$ B) $e ^ { 10 }$