Step 1: Write down the limit you want to find: lim (x -> 1) (x^3 - 1)/(x - 1).
Step 2: Notice that both the numerator (x^3 - 1) and the denominator (x - 1) equal 0 when x = 1. This means we have a 0/0 form, which we need to simplify.
Step 3: Factor the numerator (x^3 - 1). It can be factored as (x - 1)(x^2 + x + 1).
Step 4: Rewrite the limit using the factored form: lim (x -> 1) [(x - 1)(x^2 + x + 1)/(x - 1)].
Step 5: Cancel the (x - 1) in the numerator and denominator. This gives us lim (x -> 1) (x^2 + x + 1).
Step 6: Now, substitute x = 1 into the simplified expression (x^2 + x + 1): 1^2 + 1 + 1 = 3.
Step 7: Therefore, the limit is 3.
Limit Evaluation – Understanding how to evaluate limits, particularly when direct substitution leads to an indeterminate form.
Factoring – The ability to factor polynomials to simplify expressions before taking limits.
Cancellation of Terms – Recognizing when and how to cancel common factors in a limit expression.
