To evaluate the limit [tex]\(\lim _{x \rightarrow 1}\left(\frac{x-1}{x^2-1}\right)\)[/tex], we will follow a step-by-step approach:
1. Identify the Expression: The given limit is [tex]\(\lim _{x \rightarrow 1}\left(\frac{x-1}{x^2-1}\right)\)[/tex].
2. Simplify the Denominator: Notice that the denominator [tex]\(x^2 - 1\)[/tex] can be factored using the difference of squares:
[tex]\[
x^2 - 1 = (x - 1)(x + 1)
\][/tex]
3. Rewrite the Expression: Substitute the factored form of the denominator into the limit expression:
[tex]\[
\frac{x-1}{x^2-1} = \frac{x-1}{(x-1)(x+1)}
\][/tex]
4. Simplify the Fraction: Cancel the common factor [tex]\((x-1)\)[/tex] in the numerator and the denominator (note that [tex]\(x \neq 1\)[/tex] to avoid division by zero):
[tex]\[
\frac{x-1}{(x-1)(x+1)} = \frac{1}{x+1} \quad (\text{for } x \neq 1)
\][/tex]
5. Evaluate the Simplified Limit: Now, we evaluate the limit of the simplified expression [tex]\(\frac{1}{x+1}\)[/tex] as [tex]\(x\)[/tex] approaches 1:
[tex]\[
\lim _{x \rightarrow 1} \frac{1}{x+1} = \frac{1}{1+1} = \frac{1}{2}
\][/tex]
Therefore, the limit is:
[tex]\[
\lim _{x \rightarrow 1}\left(\frac{x-1}{x^2-1}\right) = \frac{1}{2}
\][/tex]
The result of the limit as [tex]\(x\)[/tex] approaches 1 is [tex]\( \frac{1}{2} \)[/tex].
