Questions: Determine [ lim x rightarrow 11 fracsin (x)-sin (11)x-11 ]

$Determine [ lim x rightarrow 11 fracsin (x)-sin (11)x-11 ]$

Transcript text: Determine \[ \lim _{x \rightarrow 11} \frac{\sin (x)-\sin (11)}{x-11} \]

Solution

Solution Steps

To solve this limit problem, we can use L'Hôpital's Rule, which is applicable when the limit results in an indeterminate form like 0/0. According to L'Hôpital's Rule, we take the derivative of the numerator and the derivative of the denominator separately and then compute the limit again.

Solution Approach

Identify that the limit is in the indeterminate form 0/0.
Apply L'Hôpital's Rule by differentiating the numerator and the denominator.
Evaluate the limit of the resulting expression as $ x $ approaches 11.

Step 1: Identify the Indeterminate Form

The given limit is

\[ \lim _{x \rightarrow 11} \frac{\sin (x)-\sin (11)}{x-11} \]

Substituting $ x = 11 $ directly into the expression results in the indeterminate form $\frac{0}{0}$.

Step 2: Apply L'Hôpital's Rule

To resolve the indeterminate form, we apply L'Hôpital's Rule, which involves differentiating the numerator and the denominator separately.

The derivative of the numerator $\sin(x) - \sin(11)$ with respect to $x$ is $\cos(x)$.
The derivative of the denominator $x - 11$ with respect to $x$ is $1$.

Step 3: Evaluate the Limit

Now, we evaluate the limit of the new expression:

\[ \lim _{x \rightarrow 11} \frac{\cos(x)}{1} = \cos(11) \]