To evaluate the limit [tex]\(\lim_{x \rightarrow 0} \frac{1 - \cos(9x)}{x^2}\)[/tex], we'll proceed with the following steps:
1. Understand the given limit: We need to determine the behavior of the function [tex]\(\frac{1 - \cos(9x)}{x^2}\)[/tex] as [tex]\(x\)[/tex] approaches 0.
2. Apply the Taylor series expansion for [tex]\(\cos(9x)\)[/tex]: Around [tex]\(x = 0\)[/tex], the cosine function can be approximated using the Taylor series expansion:
[tex]\[
\cos(9x) \approx 1 - \frac{(9x)^2}{2!} + \frac{(9x)^4}{4!} - \cdots
\][/tex]
For small values of [tex]\(x\)[/tex], higher-order terms become negligible. Therefore, we can use the truncated series:
[tex]\[
\cos(9x) \approx 1 - \frac{(9x)^2}{2}
\][/tex]
3. Substitute the approximation into the limit expression: Substitute [tex]\(\cos(9x)\)[/tex] by its approximation:
[tex]\[
\frac{1 - \cos(9x)}{x^2} \approx \frac{1 - \left(1 - \frac{(9x)^2}{2}\right)}{x^2}
\][/tex]
4. Simplify the expression: Simplify the numerator:
[tex]\[
\frac{1 - 1 + \frac{(9x)^2}{2}}{x^2} = \frac{\frac{81x^2}{2}}{x^2}
\][/tex]
5. Divide and simplify further: Cancel out [tex]\(x^2\)[/tex] in the numerator and the denominator:
[tex]\[
\frac{81x^2}{2x^2} = \frac{81}{2}
\][/tex]
6. Evaluate the resulting limit: The fraction [tex]\(\frac{81}{2}\)[/tex] does not depend on [tex]\(x\)[/tex], so it remains constant as [tex]\(x\)[/tex] approaches 0. Thus, the evaluated limit is:
[tex]\[
\lim _{x \rightarrow 0} \frac{1 - \cos 9x}{x^2} = \frac{81}{2}
\][/tex]
Therefore, the limit is:
[tex]\[
\boxed{\frac{81}{2}}
\][/tex]
