Find the limit: lim (x -> 1) (x^4 - 1)/(x - 1) (2023)

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Question: Find the limit: lim (x -> 1) (x^4 - 1)/(x - 1) (2023)

Options:

Correct Answer: 0

Exam Year: 2023

Solution:

Factoring gives ((x - 1)(x^3 + x^2 + x + 1))/(x - 1). For x ≠ 1, this simplifies to x^3 + x^2 + x + 1. Thus, lim (x -> 1) = 4.

Find the limit: lim (x -> 1) (x^4 - 1)/(x - 1) (2023)

Find the limit: lim (x -> 1) (x^4 - 1)/(x - 1) (2023)

Step 1: Identify the limit we want to find: lim (x -> 1) (x^4 - 1)/(x - 1).
Step 2: Notice that both the numerator (x^4 - 1) and the denominator (x - 1) equal 0 when x = 1. This means we can factor the numerator.
Step 3: Factor the numerator x^4 - 1. It can be factored as (x^2 - 1)(x^2 + 1).
Step 4: Further factor (x^2 - 1) into (x - 1)(x + 1). So, we have: (x - 1)(x + 1)(x^2 + 1).
Step 5: Now rewrite the limit: lim (x -> 1) ((x - 1)(x + 1)(x^2 + 1))/(x - 1).
Step 6: Since x ≠ 1, we can cancel (x - 1) from the numerator and denominator.
Step 7: This simplifies our limit to lim (x -> 1) (x + 1)(x^2 + 1).
Step 8: Now substitute x = 1 into the simplified expression: (1 + 1)(1^2 + 1) = 2 * 2.
Step 9: Calculate the result: 2 * 2 = 4.
Step 10: Therefore, the limit is 4.

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 – Recognizing when to apply L'Hôpital's Rule for limits that result in indeterminate forms.