What is the limit: lim (x -> 1) (x^2 - 1)/(x - 1)? (2019)

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Question: What is the limit: lim (x -> 1) (x^2 - 1)/(x - 1)? (2019)

Options:

Correct Answer: 2

Exam Year: 2019

Solution:

Factoring gives (x - 1)(x + 1)/(x - 1), which simplifies to x + 1. Thus, lim (x -> 1) (x + 1) = 2.

What is the limit: lim (x -> 1) (x^2 - 1)/(x - 1)? (2019)

What is the limit: lim (x -> 1) (x^2 - 1)/(x - 1)? (2019)

Step 1: Identify the limit we want to find: lim (x -> 1) (x^2 - 1)/(x - 1).
Step 2: Notice that both the numerator (x^2 - 1) and the denominator (x - 1) can be simplified.
Step 3: Factor the numerator: x^2 - 1 can be factored as (x - 1)(x + 1).
Step 4: Rewrite the limit using the factored form: lim (x -> 1) ((x - 1)(x + 1))/(x - 1).
Step 5: Cancel the (x - 1) in the numerator and denominator, as long as x is not equal to 1: lim (x -> 1) (x + 1).
Step 6: Now, substitute x = 1 into the simplified expression: (1 + 1) = 2.
Step 7: Conclude that the limit is 2.

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