Step 1: Identify the limit we want to find: lim (x -> 1) (x^2 - 1)/(x - 1)^2.
Step 2: Notice that the expression (x^2 - 1) can be factored. It factors to (x - 1)(x + 1).
Step 3: Rewrite the limit using the factored form: lim (x -> 1) ((x - 1)(x + 1))/(x - 1)^2.
Step 4: Simplify the expression by canceling one (x - 1) from the numerator and the denominator: lim (x -> 1) (x + 1)/(x - 1).
Step 5: Now, substitute x = 1 into the simplified expression: (1 + 1)/(1 - 1).
Step 6: Notice that substituting gives us 2/0, which means we need to analyze the limit further.
Step 7: As x approaches 1, the expression (x + 1) approaches 2, and (x - 1) approaches 0.
Step 8: Since (x - 1) is approaching 0, we can say that the limit approaches infinity or negative infinity depending on the direction from which x approaches 1.
Step 9: However, we can also consider the left-hand limit and right-hand limit separately to confirm the behavior.
Step 10: Conclude that the limit does not exist in the traditional sense, but we can say it approaches infinity.
Limits – Understanding how to evaluate limits, especially when direct substitution leads to indeterminate forms.
Factoring – The ability to factor expressions to simplify limits and resolve indeterminate forms.
Cancellation – Recognizing when and how to cancel common factors in a limit expression.
