Find the following limits or state that they do not exist. Assume a, b, c, and k

Question

Find the following limits or state that they do not exist. Assume a, b, c, and k are fixed real numbers.

lim_{x \to 9} \frac{\sqrt{x} - 3}{x - 9}

Answer 1

To find

lim_{x \to 9} \frac{\sqrt{x} - 3}{x - 9}

We rationalize the numerator as

lim_{x \to 9} \frac{\sqrt{x} - 3}{x - 9} \times \frac{\sqrt{x} + 3}{\sqrt{x} + 3}

= lim_{x \to 9} \frac{x + 3 \sqrt{x} - 3 \sqrt{x} - 9}{(x - 9) (\sqrt{x} + 3)}

= lim_{x \to 9} \frac{x - 9}{(x - 9) (\sqrt{x} + 3)}

= lim_{x \to 9} \frac{1}{\sqrt{x} + 3}

= \frac{1}{\sqrt{9} + 3}

= \frac{1}{6}

Therefore,

lim_{x \to 9} \frac{\sqrt{x} - 3}{x - 9} = \frac{1}{6}

Answer 2

Jeffrey Jordon

Answered 2022-07-04 Author has 2313 answers

Answer is given below (on video)

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