Find the limit: lim (x -> 1) (x^2 - 1)/(x - 1)

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Question: Find the limit: lim (x -> 1) (x^2 - 1)/(x - 1)

Options:

Correct Answer: 2

Solution:

This is an indeterminate form (0/0). Factor the numerator: (x-1)(x+1)/(x-1) = x + 1. Thus, lim (x -> 1) (x + 1) = 2.

Find the limit: lim (x -> 1) (x^2 - 1)/(x - 1)

Find the limit: lim (x -> 1) (x^2 - 1)/(x - 1)

Step 1: Identify the limit we want to find: lim (x -> 1) (x^2 - 1)/(x - 1).
Step 2: Substitute x = 1 into the expression. This gives (1^2 - 1)/(1 - 1) = 0/0, which is an indeterminate form.
Step 3: Factor the numerator x^2 - 1. It 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) terms in the numerator and denominator, as long as x is not equal to 1. This simplifies to 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.

Limits and Indeterminate Forms – Understanding how to evaluate limits, especially when encountering indeterminate forms like 0/0.
Factoring Polynomials – The ability to factor expressions to simplify limits.
Continuity of Functions – Recognizing that limits can be evaluated by substituting values into simplified functions.