Step 1: Identify the limit we want to calculate: lim (x -> 1) (x^2 - 1)/(x - 1)^2.
Step 2: Notice that both the numerator and denominator become 0 when x = 1. This means we can simplify the expression.
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: Now substitute x = 1 into the simplified expression (x + 1)/(x - 1). This gives us (1 + 1)/(1 - 1), which is 2/0.
Step 7: Since we cannot divide by zero, we need to analyze the limit further. As x approaches 1, (x + 1) approaches 2 and (x - 1) approaches 0.
Step 8: The limit approaches positive or negative infinity depending on the direction from which x approaches 1. However, since we are looking for the limit as x approaches 1, we can conclude that the limit is undefined.
Limit Calculation – Understanding how to evaluate limits, especially when dealing with indeterminate forms.
Factoring – The ability to factor expressions to simplify limits.
Indeterminate Forms – Recognizing and resolving indeterminate forms like 0/0.
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