$\text { Evaluate the limit } \lim _ { x \rightarrow \infty } \frac { x ^ { 5 } } { e ^ { 4 x } } \text { using L'Hopital's Rule if necessary. }$
A) 0
B) $\frac { 15 } { 128 }$
C) $\frac { 5 } { 4 }$
D) 1
E) does not exist

Question

$\text { Evaluate the limit } \lim _ { x \rightarrow \infty } \frac { x ^ { 5 } } { e ^ { 4 x } } \text { using L'Hopital's Rule if necessary. }$ 
A) 0
B) $\frac { 15 } { 128 }$
C) $\frac { 5 } { 4 }$
D) 1
E) does not exist

Quizplus · Accepted Answer

To evaluate the limit, we can use the fact that exponential functions grow much faster than polynomial functions. As x approaches infinity, the denominator e^(4x) grows much faster than the numerator x^5 because the base e is bigger than 1. Therefore, the fraction approaches zero, and the answer is A. L'Hopital's Rule is not necessary in this case.