To solve the limit [tex]\(\lim _{x \rightarrow \sqrt{3}} \frac{1}{x^2}\)[/tex], follow these steps:
1. Identify the Function and Point of Interest: We need to evaluate the limit of the function [tex]\(\frac{1}{x^2}\)[/tex] as [tex]\(x\)[/tex] approaches [tex]\(\sqrt{3}\)[/tex].
2. Substitute the Point into the Function: Substitute [tex]\(x = \sqrt{3}\)[/tex] directly into the function [tex]\(\frac{1}{x^2}\)[/tex].
[tex]\[
\frac{1}{(\sqrt{3})^2}
\][/tex]
3. Simplify the Expression: Simplify the exponent and the fraction. The square of [tex]\(\sqrt{3}\)[/tex] is 3.
[tex]\[
(\sqrt{3})^2 = 3
\][/tex]
4. Evaluate the Fraction: Now, the expression becomes:
[tex]\[
\frac{1}{3}
\][/tex]
Thus, the limit of the function [tex]\(\frac{1}{x^2}\)[/tex] as [tex]\(x\)[/tex] approaches [tex]\(\sqrt{3}\)[/tex] is [tex]\(\frac{1}{3}\)[/tex].
So, the final answer is:
[tex]\[
\lim _{x \rightarrow \sqrt{3}} \frac{1}{x^2} = \frac{1}{3}
\][/tex]
