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Calculus Level 3

lim x 2 x 2 x 2 2 x 2 + 4 3 = ? \large\displaystyle\lim_{x\to2^-}\dfrac{|x^2-x-2|}{2-\sqrt[3]{x^2+4}}=\, ?


The answer is 9.

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1 solution

Let x = 2 δ \displaystyle x=2-\delta where δ \displaystyle \delta is a positive real number. then the limit is x 2 x 2 2 ( x 2 + 4 ) 1 / 3 \displaystyle \frac{|x^{2}-x-2|}{2-(x^{2}+4)^{1/3}} x 2 x + 1 2 ( 8 4 δ ) 1 / 3 = x 2 x + 1 2 2 ( 1 1 6 δ ) = δ x + 1 δ 3 \displaystyle \frac{|x-2||x+1|}{2-(8-4\delta)^{1/3}}=\frac{|x-2||x+1|}{2-2(1-\frac{1}{6}\delta)}=\frac{\delta|x+1|}{\frac{\delta}{3}}

lim x > 2 = 3 x + 1 = 9 \displaystyle \lim_{x->2^{-}}=3|x+1|=9

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