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AP Calculus BC 1.6 Determining Limits Using Algebraic Manipulation– Exam Style questions with Answer- FRQ

Question 

The function is defined by \( f(x)= \frac{x^{2}-4}{x-2} \) for \(x \neq 2\).

(a) Find .

(b) Is continuous at ? Justify your answer.

(c) The function is defined by
.
Find the value of such that is continuous at .

▶️Answer/Explanation

(a) We can factor the numerator and simplify the expression:
\(\displaystyle \lim_{x \to 2}f(x) = \displaystyle \lim_{x \to 2}\frac{x^{2}-4}{x-2} =\displaystyle \lim_{x \to 2}\frac{(x+2)(x-2)}{(x-2)}=\displaystyle \lim_{x \to 2}(x+2) = 4\)

(b) No, is not continuous at because is undefined.

(c) For to be continuous at , we need .
From part (a), we know that .
Therefore, we must have .

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