Sure, let's work through the problem step by step.
We are asked to find the limit:
[tex]\[
\lim_{x \to \pi} \frac{1 - \sin x}{x}
\][/tex]
To evaluate this, let’s follow these steps:
1. Understanding the function at the limit point:
We need to find the limit as [tex]\( x \)[/tex] approaches [tex]\(\pi\)[/tex] for the function [tex]\(\frac{1 - \sin x}{x}\)[/tex]. When [tex]\( x = \pi \)[/tex],
[tex]\[
\sin(\pi) = 0,
\][/tex]
so the function simplifies to:
[tex]\[
\frac{1 - 0}{\pi} = \frac{1}{\pi}.
\][/tex]
2. Evaluating the limit:
For limits involving trigonometric functions, it's useful to remember key values and properties. At [tex]\( x = \pi \)[/tex],
[tex]\[
\sin(\pi) = 0.
\][/tex]
As [tex]\( x \)[/tex] gets infinitely close to [tex]\( \pi \)[/tex], [tex]\(\sin x\)[/tex] gets infinitely close to [tex]\( 0\)[/tex]. Therefore, the expression [tex]\(1 - \sin x\)[/tex] approaches [tex]\(1 - 0 = 1\)[/tex].
Since [tex]\(x\)[/tex] approaches [tex]\(\pi\)[/tex], the denominator simply approaches [tex]\(\pi\)[/tex].
3. Putting it all together:
Since both the numerator and denominator are approaching constants,
[tex]\[
\lim_{x \to \pi} \frac{1 - \sin x}{x} = \frac{1 - \sin(\pi)}{\pi} = \frac{1 - 0}{\pi} = \frac{1}{\pi}.
\][/tex]
Thus, the limit is:
[tex]\[
\boxed{\frac{1}{\pi}}
\][/tex]
