Evaluate the limit lim (x -> 3) (x^2 - 9)/(x - 3). Is the function continuous at x = 3? (2021)

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Q: Evaluate the limit lim (x -> 3) (x^2 - 9)/(x - 3). Is the function continuous at x = 3? (2021)

Solution: lim (x -> 3) (x^2 - 9)/(x - 3) = lim (x -> 3) (x + 3) = 6. The function is not defined at x = 3, hence not continuous.

Steps: 9

Step 1: Identify the limit we need to evaluate: lim (x -> 3) (x^2 - 9)/(x - 3).
Step 2: Notice that when we substitute x = 3 directly into the expression, we get (3^2 - 9)/(3 - 3) = (9 - 9)/(0) = 0/0, which is undefined.
Step 3: To resolve the 0/0 form, we can factor the numerator: x^2 - 9 is a difference of squares, which factors to (x - 3)(x + 3).
Step 4: Rewrite the limit using the factored form: lim (x -> 3) ((x - 3)(x + 3))/(x - 3).
Step 5: Cancel the (x - 3) terms in the numerator and denominator, but note that this is valid only for x ≠ 3: lim (x -> 3) (x + 3).
Step 6: Now substitute x = 3 into the simplified expression: 3 + 3 = 6.
Step 7: Conclude that lim (x -> 3) (x^2 - 9)/(x - 3) = 6.
Step 8: Check if the function is continuous at x = 3. The function is not defined at x = 3 because the original expression has a denominator of 0.
Step 9: Since the function is not defined at x = 3, it is not continuous at that point.