Question

it: $lim_{x \to 13} \frac{x - 13}{sqrt{x + 3} - 4}$

Explanation:

Step1: Rationalize the denominator

Multiply the numerator and denominator by the conjugate of the denominator \(\sqrt{x + 3}+4\).

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Using the difference - of - squares formula \((a - b)(a + b)=a^{2}-b^{2}\), where \(a=\sqrt{x + 3}\) and \(b = 4\), the denominator becomes \((\sqrt{x+3})^{2}-4^{2}=x + 3-16=x - 13\).
So the limit becomes \(\lim_{x
ightarrow13}\frac{(x - 13)(\sqrt{x + 3}+4)}{x - 13}\).

Step2: Simplify the expression

Cancel out the common factor \((x - 13)\) (since \(x
eq13\) when taking the limit, we can do this cancellation).

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Step3: Evaluate the limit

Substitute \(x = 13\) into \(\sqrt{x + 3}+4\).

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Answer:

\(8\)