Question

question
evaluate the limit: \\( \lim_{x \to 6} \frac{-4x + 24}{\sqrt{x + 10} - 4} \\)

answer attempt 1 out of 2

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Explanation:

Step1: Simplify the numerator

Factor out -4 from the numerator: $-4x + 24 = -4(x - 6)$

Step2: Rationalize the denominator

Multiply the numerator and denominator by the conjugate of the denominator, which is $\sqrt{x + 10} + 4$:
\[

$$\begin{align*} &\lim_{x\to 6} \frac{-4(x - 6)}{\sqrt{x + 10} - 4} \cdot \frac{\sqrt{x + 10} + 4}{\sqrt{x + 10} + 4}\\ =&\lim_{x\to 6} \frac{-4(x - 6)(\sqrt{x + 10} + 4)}{(x + 10) - 16}\\ =&\lim_{x\to 6} \frac{-4(x - 6)(\sqrt{x + 10} + 4)}{x - 6} \end{align*}$$

\]

Step3: Cancel out the common factor

Cancel out the $(x - 6)$ terms (since $x\to 6$ but $x
eq 6$ at the limit point, we can do this):
\[
\lim_{x\to 6} -4(\sqrt{x + 10} + 4)
\]

Step4: Substitute x = 6

Substitute $x = 6$ into the expression:
\[
-4(\sqrt{6 + 10} + 4) = -4(\sqrt{16} + 4) = -4(4 + 4) = -4\times 8 = -32
\]

Answer:

-32