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Evaluating a 0/0 limit using L'Hopital's Rule

Evaluate the limit lim⁡x→4x−2x−4\lim_{x \to 4} \frac{\sqrt{x} - 2}{x - 4} using L'Hopital's Rule. Be sure to verify that the limit is in an indeterminate form before applying the rule.

Hint

Check that plugging in x=4x = 4 gives 0/00/0. If so, differentiate the numerator and denominator separately (not using the quotient rule), then substitute x=4x = 4 into the resulting expression.

Step-by-step solution

Substituting x=4x = 4 gives 4−24−4=00\frac{\sqrt{4}-2}{4-4} = \frac{0}{0}, which confirms the indeterminate form, so L'Hopital's Rule applies. Differentiate the numerator: ddx(x−2)=12x\frac{d}{dx}(\sqrt{x}-2) = \frac{1}{2\sqrt{x}}. Differentiate the denominator: ddx(x−4)=1\frac{d}{dx}(x-4) = 1.

By L'Hopital's Rule, lim⁡x→4x−2x−4=lim⁡x→412x1=124.\lim_{x\to 4}\frac{\sqrt{x}-2}{x-4} = \lim_{x\to 4}\frac{\frac{1}{2\sqrt{x}}}{1} = \frac{1}{2\sqrt{4}}. Evaluating gives 14\frac{1}{4}, which is the value of the original limit.

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