To find the limit \(\lim_{x \rightarrow \theta} \frac{x \sin \theta - \theta \sin x}{x - \theta}\), we can proceed by carefully analyzing and simplifying the expression step by step.
1. Define the limit expression:
[tex]\[
L = \lim_{x \rightarrow \theta} \frac{x \sin \theta - \theta \sin x}{x - \theta}
\][/tex]
2. Recognize that direct substitution of \(x = \theta\) leads to an indeterminate form \( \frac{0}{0} \).
This suggests that we might need to use L'Hospital's Rule, which is applied to evaluate limits of the form \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). L'Hospital's Rule states that:
[tex]\[
\lim_{x \rightarrow \theta} \frac{f(x)}{g(x)} = \lim_{x \rightarrow \theta} \frac{f'(x)}{g'(x)}
\][/tex]
if the limit exists.
3. Identify \(f(x)\) and \(g(x)\) from the expression:
Here,
[tex]\[
f(x) = x \sin \theta - \theta \sin x
\][/tex]
[tex]\[
g(x) = x - \theta
\][/tex]
4. Differentiate \(f(x)\) and \(g(x)\) with respect to \(x\):
The derivative of \(f(x)\) with respect to \(x\) is:
[tex]\[
f'(x) = \frac{d}{dx} \left( x \sin \theta - \theta \sin x \right) = \sin \theta - \theta \cos x
\][/tex]
The derivative of \(g(x)\) with respect to \(x\) is:
[tex]\[
g'(x) = \frac{d}{dx} \left( x - \theta \right) = 1
\][/tex]
5. Apply L'Hospital's Rule:
[tex]\[
L = \lim_{x \rightarrow \theta} \frac{\sin \theta - \theta \cos x}{1}
\][/tex]
6. Simplify the expression under the limit:
[tex]\[
L = \lim_{x \rightarrow \theta} (\sin \theta - \theta \cos x)
\][/tex]
7. Evaluate the limit as \(x\) approaches \(\theta\):
By direct substitution, we have:
[tex]\[
L = \sin \theta - \theta \cos \theta
\][/tex]
Therefore, the limit is:
[tex]\[
\boxed{-\theta \cos \theta + \sin \theta}
\][/tex]
