Calculate the limit: lim (x -> 1) (x^2 - 1)/(x - 1)

Calculate the limit: lim (x -> 1) (x^2 - 1)/(x - 1)

Step 1: Identify the limit we need to calculate: lim (x -> 1) (x^2 - 1)/(x - 1).
Step 2: Substitute x = 1 into the expression. This gives us (1^2 - 1)/(1 - 1) = (1 - 1)/(0) = 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).
Step 5: Cancel the (x - 1) terms in the numerator and denominator, as long as x is not equal to 1. This simplifies to lim (x -> 1) (x + 1).
Step 6: Now substitute x = 1 into the simplified expression: 1 + 1 = 2.
Step 7: Therefore, the limit is 2.

Limits and Indeterminate Forms – Understanding how to evaluate limits, particularly when encountering indeterminate forms like 0/0.
Factoring Polynomials – The ability to factor expressions to simplify limits and resolve indeterminate forms.