Determine the limit: lim (x -> 1) (x^3 - 1)/(x - 1) (2020)

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Question: Determine the limit: lim (x -> 1) (x^3 - 1)/(x - 1) (2020)

Options:

Correct Answer: 3

Exam Year: 2020

Solution:

Factoring gives (x - 1)(x^2 + x + 1)/(x - 1). For x ≠ 1, this simplifies to x^2 + x + 1. Evaluating at x = 1 gives 3.

Determine the limit: lim (x -> 1) (x^3 - 1)/(x - 1) (2020)

Determine the limit: lim (x -> 1) (x^3 - 1)/(x - 1) (2020)

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 can 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: Since x ≠ 1, we can cancel (x - 1) from 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). This gives us 1^2 + 1 + 1 = 3.
Step 7: Therefore, the limit is 3.

Limits – Understanding how to evaluate limits, especially when direct substitution leads to indeterminate forms.
Factoring – The ability to factor polynomials to simplify expressions before evaluating limits.
Continuity – Recognizing that limits can be evaluated by simplifying expressions that are continuous at the point of interest.