Step 1: Identify the limit we want to evaluate: lim (x -> 1) (x^2 - 1)/(x - 1).
Step 2: Substitute x = 1 into the expression. We get (1^2 - 1)/(1 - 1) = 0/0, which is an indeterminate form.
Step 3: Since we have an indeterminate form, we can use L'Hôpital's Rule. This rule states that if we have 0/0 or ∞/∞, we can take the derivative of the numerator and the derivative of the denominator.
Step 4: Find the derivative of the numerator (x^2 - 1). The derivative is 2x.
Step 5: Find the derivative of the denominator (x - 1). The derivative is 1.
Step 6: Now we can rewrite the limit using the derivatives: lim (x -> 1) (2x)/(1).
Step 7: Substitute x = 1 into the new expression: (2*1)/(1) = 2/1 = 2.
Step 8: Therefore, the limit evaluates to 2.
Limits and Continuity – Understanding how to evaluate limits, especially when they result in indeterminate forms.
L'Hôpital's Rule – Applying L'Hôpital's Rule to resolve indeterminate forms like 0/0.
Factoring Polynomials – Recognizing that the expression can be simplified by factoring before taking the limit.
