Find the limit: lim (x -> 3) (x^2 - 9)/(x - 3) (2023)

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Question: Find the limit: lim (x -> 3) (x^2 - 9)/(x - 3) (2023)

Options:

Correct Answer: 0

Exam Year: 2023

Solution:

The expression can be factored as ((x - 3)(x + 3))/(x - 3). For x ≠ 3, this simplifies to x + 3. Thus, lim (x -> 3) (x + 3) = 6.

Find the limit: lim (x -> 3) (x^2 - 9)/(x - 3) (2023)

Find the limit: lim (x -> 3) (x^2 - 9)/(x - 3) (2023)

Step 1: Identify the limit we need to find: 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) as (x - 3)(x + 3).
Step 4: Rewrite the limit using the factored form: lim (x -> 3) ((x - 3)(x + 3))/(x - 3).
Step 5: For x ≠ 3, we can cancel (x - 3) in the numerator and denominator.
Step 6: This simplifies the expression to lim (x -> 3) (x + 3).
Step 7: Now, substitute x = 3 into the simplified expression: 3 + 3 = 6.
Step 8: Therefore, the limit is 6.

Limit Evaluation – Understanding how to evaluate limits, especially when dealing with indeterminate forms.
Factoring – The ability to factor expressions to simplify limits.
Continuity – Recognizing that the limit can be evaluated by substituting values when the function is continuous.