Which of the following is true for the limit $\lim _ { x \rightarrow 0 } \left( \frac { x e ^ { 2 x } } { \sin x } \right)$ ?

A)-3
B)-1
C)0
D)1
E)3

Question

Which of the following is true for the limit $\lim _ { x \rightarrow 0 } \left( \frac { x e ^ { 2 x } } { \sin x } \right)$ ?

A)-3
B)-1
C)0
D)1
E)3

Quizplus · Accepted Answer

The limit can be evaluated using L'Hôpital's Rule, which applies because the function forms an indeterminate form of 0/0 as x approaches 0. Differentiating the numerator and the denominator separately with respect to x gives: $\lim _ { x \rightarrow 0 } \left( \frac { e ^ { 2 x } + 2x e ^ { 2 x } } { \cos x } \right)$. Evaluating this at x = 0 gives $ \frac { 1 + 0 } { 1 } = 1$.

Which of the Following Is True for the Limit lim⁡x→0(xe2xsin⁡x)\lim _ { x \rightarrow 0 } \left( \frac { x e ^ { 2 x } } { \sin x } \right)limx→0​(sinxxe2x​)

Which of the Following Is True for the Limit $\lim _ { x \rightarrow 0 } \left( \frac { x e ^ { 2 x } } { \sin x } \right)$