To find the limit of the expression [tex]\(\frac{x^2 - 1}{|x - 1|}\)[/tex] as [tex]\(x\)[/tex] approaches 1 from the left ([tex]\(x \to 1^-\)[/tex]), we need to carefully analyze the behavior of the numerator and the denominator around [tex]\(x = 1\)[/tex].
1. Rewrite the Expression:
[tex]\[
\frac{x^2 - 1}{|x - 1|}
\][/tex]
2. Factorize the Numerator:
The term [tex]\(x^2 - 1\)[/tex] can be factorized using the difference of squares:
[tex]\[
x^2 - 1 = (x - 1)(x + 1)
\][/tex]
Therefore, the expression becomes:
[tex]\[
\frac{(x - 1)(x + 1)}{|x - 1|}
\][/tex]
3. Consider the Absolute Value:
The absolute value [tex]\(|x - 1|\)[/tex] depends on whether [tex]\(x - 1\)[/tex] is positive or negative. As we are interested in the limit as [tex]\(x\)[/tex] approaches 1 from the left ([tex]\(x \to 1^-\)[/tex]), [tex]\(x - 1\)[/tex] is negative. Hence, [tex]\(|x - 1| = -(x - 1)\)[/tex].
4. Substitute for [tex]\(|x - 1|\)[/tex]:
Replacing [tex]\(|x - 1|\)[/tex] with [tex]\(-(x - 1)\)[/tex] in our expression, we get:
[tex]\[
\frac{(x - 1)(x + 1)}{-(x - 1)}
\][/tex]
5. Simplify the Expression:
The [tex]\((x - 1)\)[/tex] terms cancel each other out, leaving us with:
[tex]\[
\frac{(x - 1)(x + 1)}{-(x - 1)} = \frac{x + 1}{-1} = -(x + 1)
\][/tex]
6. Take the Limit as [tex]\(x \to 1^-\)[/tex]:
Now, we need to evaluate the expression [tex]\(-(x + 1)\)[/tex] as [tex]\(x\)[/tex] approaches 1 from the left:
[tex]\[
\lim_{x \to 1^-} -(x + 1) = -((1) + 1) = -2
\][/tex]
Therefore, the limit of the given expression as [tex]\(x\)[/tex] approaches 1 from the left is:
[tex]\[
\boxed{-2}
\][/tex]
