Question

prove that

Lim x->4 ((x-4)/(2 -(x)^1/2)) = -4

Answer 1

The easiest method is to use L'hopital's rule:\[\lim_{x \rightarrow 4} \frac{x-4}{2-\sqrt{x}} = \lim_{x \rightarrow 4} \frac{1}{-\frac{1}{2}x^\frac{-1}{2}} = \lim_{x \rightarrow 4}-2\sqrt{x} = -4 \]The second, slightly longer method is to factor out the numerator as a difference of squares.\[\lim_{x \rightarrow 4} \frac{x-4}{2-\sqrt{x}} = \lim_{x \rightarrow 4} \frac{(\sqrt{x} - 2)(\sqrt{x}+2)}{-(\sqrt{x} - 2)} = \lim_{x \rightarrow 4} -(\sqrt{x} + 2) = -4\]

Answer 2

You Just Simply Can't Apply L'Hos.. rule to the Problem. You have to state the Criteria under which the Rule can be applied. With out the right condition, the Application of the Rule fails. All for Information and Guidance.