To solve the limit [tex]\(\lim _{x \rightarrow y} \frac{\cos x - \cos y}{x - y}\)[/tex], we need to examine the expression carefully and find a way to simplify it. The limit of a difference quotient often suggests that derivatives can be involved. Here’s a step-by-step explanation of how to reach the result:
1. Recognize the Form of the Limit:
The given expression [tex]\(\lim _{x \rightarrow y} \frac{\cos x - \cos y}{x - y}\)[/tex] resembles the definition of the derivative of a function at a point.
Recall that the derivative of a function [tex]\(f(x)\)[/tex] at a point [tex]\(a\)[/tex] is given by:
[tex]\[
f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}
\][/tex]
In our case, the expression [tex]\(\frac{\cos x - \cos y}{x - y}\)[/tex] can be viewed as the derivative of [tex]\(\cos(x)\)[/tex] evaluated at [tex]\(x = y\)[/tex].
2. Apply the Concept of Derivative:
Consider the function [tex]\(f(x) = \cos(x)\)[/tex]. The derivative of [tex]\(f(x)\)[/tex] is [tex]\(f'(x) = -\sin(x)\)[/tex].
3. Evaluate at the Point [tex]\(y\)[/tex]:
Since [tex]\(x\)[/tex] approaches [tex]\(y\)[/tex], the derivative of [tex]\(\cos(x)\)[/tex] at [tex]\(x = y\)[/tex] is:
[tex]\[
f'(y) = -\sin(y)
\][/tex]
4. Interpret the Limit:
The given limit can now be interpreted as the derivative of [tex]\(\cos(x)\)[/tex] evaluated at [tex]\(x = y\)[/tex]:
[tex]\[
\lim _{x \rightarrow y} \frac{\cos x - \cos y}{x - y} = -\sin(y)
\][/tex]
Therefore, the solution to the limit is:
[tex]\[
\boxed{-\sin(y)}
\][/tex]
