To find the limit [tex]\(\lim_{x \to 1} \frac{x^2 - 3}{(x-1)^2}\)[/tex], we will proceed with the analysis in a structured way. Let's break it down step-by-step:
1. Substitute the point [tex]\(x = 1\)[/tex] into the expression:
When we substitute [tex]\(x = 1\)[/tex] into the expression [tex]\(\frac{x^2 - 3}{(x - 1)^2}\)[/tex], we get:
[tex]\[
\frac{1^2 - 3}{(1 - 1)^2} = \frac{1 - 3}{0} = \frac{-2}{0}
\][/tex]
This indicates a form of [tex]\(\frac{-2}{0}\)[/tex].
2. Interpret the result:
Since the denominator [tex]\((x - 1)^2\)[/tex] approaches zero as [tex]\(x\)[/tex] approaches 1, we need to analyze the behavior of the numerator and denominator near [tex]\(x = 1\)[/tex].
- The numerator, [tex]\(x^2 - 3\)[/tex], approaches [tex]\(1 - 3 = -2\)[/tex] as [tex]\(x\)[/tex] approaches 1.
- The denominator, [tex]\((x - 1)^2\)[/tex], approaches [tex]\(0\)[/tex], but [tex]\((x - 1)^2\)[/tex] is always positive for all [tex]\(x \neq 1\)[/tex].
3. Conclusion about the limit:
Since the numerator is [tex]\( -2 \)[/tex] and remains constant in the vicinity of [tex]\(x = 1\)[/tex], and the denominator [tex]\((x - 1)^2\)[/tex] approaches [tex]\(0\)[/tex] but remains positive, the fraction [tex]\(\frac{-2}{(x - 1)^2}\)[/tex] rapidly decreases without bound as [tex]\(x\)[/tex] approaches 1.
4. Behavior analysis:
Therefore, as [tex]\(x\)[/tex] approaches 1, [tex]\(\frac{x^2 - 3}{(x - 1)^2}\)[/tex] becomes indefinitely large in the negative direction. Hence,
[tex]\[
\lim_{x \to 1} \frac{x^2 - 3}{(x - 1)^2} = -\infty
\][/tex]
The final result of the limit is:
[tex]\[
\boxed{-\infty}
\][/tex]
