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

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

Options:

Correct Answer: 3

Solution:

Factoring gives (x-1)(x^2 + x + 1)/(x-1) = x^2 + x + 1, thus limit is 3.

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

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

Correct Answer: 3

Step 1: Identify the limit we want to evaluate: lim x->1 of (x^3 - 1)/(x - 1).
Step 2: Notice that if we plug in x = 1 directly, both the numerator and denominator become 0, which is an 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 of [(x - 1)(x^2 + 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.
Step 6: Now we have lim x->1 of (x^2 + x + 1).
Step 7: Substitute x = 1 into the simplified expression: 1^2 + 1 + 1 = 3.
Step 8: Therefore, 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 taking limits.
L'Hôpital's Rule – An alternative method for evaluating limits that result in indeterminate forms, though not used in this solution.