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Find the limit as x approaches 4 of (sqrt(x) - 2)/(x - 4)

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Answer

\boxed{\frac{1}{4}}

Substituting values very close to $4$ into the original expression confirms that the output approaches $0.25$ .

Problem: Find the limit $\lim_{x \to 4} \frac{\sqrt{x} - 2}{x - 4}$ .

Steps

Rationalize the numerator $\lim_{x \to 4} \frac{(\sqrt{x} - 2)(\sqrt{x} + 2)}{(x - 4)(\sqrt{x} + 2)}$
Multiply out the numerator $\lim_{x \to 4} \frac{x - 4}{(x - 4)(\sqrt{x} + 2)}$
Cancel the common factor of $(x - 4)$ $\lim_{x \to 4} \frac{1}{\sqrt{x} + 2}$
Substitute $x = 4$ $\frac{1}{\sqrt{4} + 2} = \frac{1}{2 + 2}$
Simplify the fraction $\frac{1}{4}$

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