Certainly! Let's solve the limit [tex]\(\lim_{x \rightarrow \theta} \frac{x \cos \theta - \theta \cos x}{x - \theta}\)[/tex] step by step.
First, notice that directly substituting [tex]\(x = \theta\)[/tex] into the expression leads to an indeterminate form [tex]\(\frac{0}{0}\)[/tex]. This suggests that we need to use L'Hôpital's Rule, which is often helpful in evaluating limits of this type.
L'Hôpital's Rule states that if [tex]\(\lim_{x \to c} \frac{f(x)}{g(x)}\)[/tex] results in an indeterminate form [tex]\(\frac{0}{0}\)[/tex] or [tex]\(\frac{\infty}{\infty}\)[/tex], then:
[tex]\[
\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)},
\][/tex]
provided the limit on the right side exists.
Here, our [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] are:
[tex]\[ f(x) = x \cos \theta - \theta \cos x \][/tex]
[tex]\[ g(x) = x - \theta \][/tex]
To apply L'Hôpital's Rule, we need to differentiate the numerator and the denominator with respect to [tex]\(x\)[/tex].
Differentiate the numerator:
[tex]\[
f(x) = x \cos \theta - \theta \cos x
\][/tex]
Differentiating term by term:
[tex]\[
f'(x) = \frac{d}{dx}(x \cos \theta) - \frac{d}{dx}(\theta \cos x)
\][/tex]
1. [tex]\(\frac{d}{dx}(x \cos \theta) = \cos \theta\)[/tex], since [tex]\(\cos \theta\)[/tex] is a constant with respect to [tex]\(x\)[/tex].
2. [tex]\(\frac{d}{dx}(\theta \cos x) = -\theta \sin x\)[/tex], using the chain rule. Since [tex]\(\theta\)[/tex] is constant with respect to [tex]\(x\)[/tex], the derivative of [tex]\(\cos x\)[/tex] is [tex]\(-\sin x\)[/tex].
Therefore,
[tex]\[
f'(x) = \cos \theta - (-\theta \sin x) = \cos \theta + \theta \sin x
\][/tex]
Differentiate the denominator:
[tex]\[
g(x) = x - \theta
\][/tex]
Differentiating:
[tex]\[
g'(x) = 1
\][/tex]
Now, applying L'Hôpital's Rule:
[tex]\[
\lim_{x \rightarrow \theta} \frac{x \cos \theta - \theta \cos x}{x - \theta} = \lim_{x \rightarrow \theta} \frac{\cos \theta + \theta \sin x}{1}
\][/tex]
Since the denominator is now [tex]\(1\)[/tex], we simply evaluate the limit of the numerator as [tex]\(x\)[/tex] approaches [tex]\(\theta\)[/tex]:
[tex]\[
\lim_{x \rightarrow \theta} (\cos \theta + \theta \sin x) = \cos \theta + \theta \sin (\theta)
\][/tex]
Thus, our final result is:
[tex]\[
\lim_{x \rightarrow \theta} \frac{x \cos \theta - \theta \cos x}{x - \theta} = \theta \sin \theta + \cos \theta
\][/tex]
So, the detailed solution to the limit is:
[tex]\[
\theta \sin \theta + \cos \theta
\][/tex]
