Determine the value of the limit $\lim _ { x \rightarrow 0 } \frac { \ln ( \sec x ) } { x ^ { 2 } }$ ?
A) $- \infty$
B) $- \frac { 1 } { 2 }$
C) $\frac { 1 } { 2 }$
D)0
E) $\infty$

Question

Quizplus · Accepted Answer

The limit $\lim _ { x \rightarrow 0 } \frac { \ln ( \sec x ) } { x ^ { 2 } }$ can be evaluated using L'Hôpital's Rule, since it is of the form $\frac{0}{0}$ as $x$ approaches 0. Differentiating the numerator and denominator separately with respect to $x$ gives $\lim _ { x \rightarrow 0 } \frac { \sec x \tan x } { 2x }$. Applying L'Hôpital's Rule again on this form, we get $\lim _ { x \rightarrow 0 } \frac { \sec^2 x + \sec x \tan^2 x } { 2 }$, which simplifies to $\frac{1}{2}$ as $x$ approaches 0, since $\sec(0) = 1$ and $\tan(0) = 0$.

Determine the Value of the Limit lim⁡x→0ln⁡(sec⁡x)x2\lim _ { x \rightarrow 0 } \frac { \ln ( \sec x ) } { x ^ { 2 } }limx→0​x2ln(secx)​

Determine the Value of the Limit $\lim _ { x \rightarrow 0 } \frac { \ln ( \sec x ) } { x ^ { 2 } }$