Calculate the limit: lim (x -> 1) (x^3 - 1)/(x - 1)

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

Options:

Correct Answer: 3

Solution:

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

Calculate the limit: lim (x -> 1) (x^3 - 1)/(x - 1)

Calculate the limit: lim (x -> 1) (x^3 - 1)/(x - 1)

Step 1: Identify the limit you need to calculate: lim (x -> 1) (x^3 - 1)/(x - 1).
Step 2: Notice that if you plug in x = 1 directly, both the numerator and denominator become 0. This means we need to simplify the expression.
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 you lim (x -> 1) (x^2 + x + 1).
Step 6: Now, plug in x = 1 into the simplified expression: 1^2 + 1 + 1 = 3.
Step 7: Therefore, the limit is 3.

Limit Calculation – Understanding how to evaluate limits, particularly when faced with indeterminate forms like 0/0.
Factoring Polynomials – 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.