$\text { Evaluate the limit } \lim _ { x \rightarrow 0 } \frac { \arcsin ( 13 x ) } { 2 x } \text { using L'Hopital's Rule if necessary. }$
A) $\frac { 2 } { 13 }$
B) 0
C) $\frac { 1 } { 2 }$
D) $\frac { 13 } { 2 }$
E) does not exist

Question

$\text { Evaluate the limit } \lim _ { x \rightarrow 0 } \frac { \arcsin ( 13 x ) } { 2 x } \text { using L'Hopital's Rule if necessary. }$ 
A) $\frac { 2 } { 13 }$
B) 0
C) $\frac { 1 } { 2 }$
D) $\frac { 13 } { 2 }$
E) does not exist

Quizplus · Accepted Answer

To solve this limit, we can apply L'Hôpital's Rule since the limit is of the form $\frac{0}{0}$ as $x$ approaches 0. Differentiating the numerator and denominator separately with respect to $x$, we get:Numerator's derivative: $\frac{d}{dx} \arcsin(13x) = \frac{13}{\sqrt{1-(13x)^2}}$Denominator's derivative: $\frac{d}{dx} 2x = 2$Thus, the limit becomes:$\lim _ { x \rightarrow 0 } \frac { \arcsin ( 13 x ) } { 2 x } = \lim _ { x \rightarrow 0 } \frac { 13 } { 2 \sqrt{1-(13x)^2} } = \frac { 13 } { 2 \sqrt{1-0} } = \frac { 13 } { 2 }$Therefore, the correct answer is $\frac { 13 } { 2 }$.