Evaluate the limit: lim (x -> 3) (x^2 - 9)/(x - 3) (2020)

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Question: Evaluate the limit: lim (x -> 3) (x^2 - 9)/(x - 3) (2020)

Options:

Correct Answer: 6

Exam Year: 2020

Solution:

Factoring gives (x - 3)(x + 3)/(x - 3). Canceling (x - 3) gives lim (x -> 3) (x + 3) = 6.

Evaluate the limit: lim (x -> 3) (x^2 - 9)/(x - 3) (2020)

Evaluate the limit: lim (x -> 3) (x^2 - 9)/(x - 3) (2020)

Step 1: Identify the limit we need to evaluate: lim (x -> 3) (x^2 - 9)/(x - 3).
Step 2: Notice that the expression (x^2 - 9) can be factored. It is a difference of squares.
Step 3: Factor (x^2 - 9) into (x - 3)(x + 3).
Step 4: Rewrite the limit using the factored form: lim (x -> 3) ((x - 3)(x + 3))/(x - 3).
Step 5: Since (x - 3) is in both the numerator and the denominator, we can cancel it out, but only for x ≠ 3.
Step 6: After canceling, we have lim (x -> 3) (x + 3).
Step 7: Now, substitute x = 3 into (x + 3): 3 + 3 = 6.
Step 8: Therefore, the limit is 6.

Limit Evaluation – Understanding how to evaluate limits, particularly when direct substitution leads to an indeterminate form.
Factoring – Using algebraic techniques to simplify expressions before evaluating limits.
Cancellation of Terms – Recognizing when and how to cancel common factors in a limit expression.