Step 1: Identify the limit we need to evaluate: lim(x->1) (x^2 - 1)/(x - 1)^2.
Step 2: Substitute x = 1 into the expression. This gives us (1^2 - 1)/(1 - 1)^2 = (0)/(0), which is an indeterminate form.
Step 3: Factor the numerator x^2 - 1. It can be factored as (x - 1)(x + 1).
Step 4: Rewrite the limit using the factored form: lim(x->1) ((x - 1)(x + 1))/(x - 1)^2.
Step 5: Simplify the expression. The (x - 1) in the numerator and one (x - 1) in the denominator cancel out, leaving us with lim(x->1) (x + 1)/(x - 1).
Step 6: Substitute x = 1 into the simplified expression: (1 + 1)/(1 - 1) = 2/0, which is still undefined.
Step 7: Since we have a limit approaching 1, we can evaluate the limit from the left and right. As x approaches 1, (x + 1) approaches 2.
Step 8: Therefore, the limit is 2.
Limit Evaluation – Understanding how to evaluate limits, especially when encountering indeterminate forms.
Factoring – The ability to factor expressions to simplify limits.
Indeterminate Forms – Recognizing and resolving indeterminate forms like 0/0.
