Determine the limit (if it exists). $$\lim _ { x \rightarrow 0 } \frac { 12 ( 1 - \cos x ) } { x ^ { 2 } }$$
A) 6
B) 48
C) 10
D) 24
E) does not exist

Question

Quizplus · Accepted Answer

The limit can be solved by using L'Hôpital's Rule, which allows us to differentiate the numerator and the denominator separately because the original limit evaluates to an indeterminate form 0/0. Differentiating the numerator gives us $12(\sin x)$ and differentiating the denominator gives us $2x$. Applying L'Hôpital's Rule again to the resulting limit $\lim _ { x \rightarrow 0 } \frac { 12 \sin x } { 2x }$ because it still evaluates to an indeterminate form, we differentiate the numerator and denominator again. The numerator becomes $12\cos x$ and the denominator becomes 2. Evaluating this at $x = 0$ gives us $12\cos(0)/2 = 12(1)/2 = 6$, hence the limit exists and is equal to 6.

Determine the Limit (If It Exists) lim⁡x→012(1−cos⁡x)x2\lim _ { x \rightarrow 0 } \frac { 12 ( 1 - \cos x ) } { x ^ { 2 } }x→0lim​x212(1−cosx)​

Determine the Limit (If It Exists) $\lim _ { x \rightarrow 0 } \frac { 12 ( 1 - \cos x ) } { x ^ { 2 } }$