Evaluate the limit lim x->1 (x^3 - 1)/(x - 1).

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Question: Evaluate the limit lim x->1 (x^3 - 1)/(x - 1).

Options:

Correct Answer: 2

Solution:

Factoring gives (x-1)(x^2 + x + 1)/(x - 1). Canceling (x - 1) gives lim x->1 (x^2 + x + 1) = 3.

Evaluate the limit lim x->1 (x^3 - 1)/(x - 1).

Evaluate the limit lim x->1 (x^3 - 1)/(x - 1).

Correct Answer: 3

Step 1: Identify the limit you need to evaluate: lim x->1 (x^3 - 1)/(x - 1).
Step 2: Notice that both the numerator (x^3 - 1) and the denominator (x - 1) become 0 when x = 1. This means we have a 0/0 indeterminate form.
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 simplifies the limit to lim x->1 (x^2 + x + 1).
Step 6: Now substitute x = 1 into the simplified expression: (1^2 + 1 + 1) = 3.
Step 7: Conclude that the limit is 3.

Limit Evaluation – Understanding how to evaluate limits, particularly when direct substitution leads to an indeterminate form.
Factoring Polynomials – The ability to factor polynomials to simplify expressions before evaluating limits.
Canceling Terms – Recognizing when and how to cancel common factors in a limit expression.