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

Question

Quizplus · Accepted Answer

The given limit can be simplified by finding a common denominator and simplifying the expression, which leads to $\lim _ { x \rightarrow 0 } \left( \frac { \cos x - 1 } { x ^ { 2 } \cos x } \right)$. Using the Taylor series expansion for $\cos x$ around 0, $\cos x \approx 1 - \frac{x^2}{2}$, the expression simplifies to $\lim _ { x \rightarrow 0 } \left( \frac { -\frac{x^2}{2} } { x ^ { 2 } \cos x } \right)$, which further simplifies to $-\frac{1}{2}$ as $x$ approaches 0, assuming $\cos x$ does not significantly deviate from 1 near 0.

Determine the Value of the Limit lim⁡x→0(1x2−1x2cos⁡x)\lim _ { x \rightarrow 0 } \left( \frac { 1 } { x ^ { 2 } } - \frac { 1 } { x ^ { 2 } \cos x } \right)limx→0​(x21​−x2cosx1​)

Determine the Value of the Limit $\lim _ { x \rightarrow 0 } \left( \frac { 1 } { x ^ { 2 } } - \frac { 1 } { x ^ { 2 } \cos x } \right)$