Determine $\lim _{x \rightarrow 0} \frac{\cos x-1}{x^{2}}$
if the limit exists.
A) -1
B) $-\frac{1}{2}$
C) $\frac{1}{2}$
D) 1
E) The limit does not exist.

Question

Determine $\lim _{x \rightarrow 0} \frac{\cos x-1}{x^{2}}$
if the limit exists.
A)  -1 
B) $-\frac{1}{2}$
C) $\frac{1}{2}$
D) 1
E) The limit does not exist.

Quizplus · Accepted Answer

The limit can be found using L'Hôpital's Rule. Applying it twice, we differentiate the numerator and the denominator separately each time. The first differentiation gives $\frac{-\sin x}{2x}$, and the second differentiation gives $\frac{-\cos x}{2}$. Evaluating this at $x = 0$ gives $\frac{-1}{2}$, hence the limit is $-\frac{1}{2}$.

Determine lim⁡x→0cos⁡x−1x2\lim _{x \rightarrow 0} \frac{\cos x-1}{x^{2}}limx→0​x2cosx−1​ If the Limit Exists −12-\frac{1}{2}−21​

Determine $\lim _{x \rightarrow 0} \frac{\cos x-1}{x^{2}}$ If the Limit Exists $-\frac{1}{2}$