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Question Number 92511 by Ar Brandon last updated on 07/May/20

lim_(x→0) (((x)^(1/3) −(x)^(1/5) )/((x)^(1/3) −(x)^(1/4) ))

$$\underset{\mathrm{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\sqrt[{\mathrm{3}}]{\mathrm{x}}−\sqrt[{\mathrm{5}}]{\mathrm{x}}}{\sqrt[{\mathrm{3}}]{\mathrm{x}}−\sqrt[{\mathrm{4}}]{\mathrm{x}}} \\ $$

Commented by john santu last updated on 07/May/20

x = t^(60) lim_(t→0) ((t^(20) −t^(12) )/(t^(20) −t^(15) )) = lim_(t→0) ((t^(12) (t^8 −1))/(t^(15) (t^5 −1))) = lim_(t→0) ((t^8 −1)/(t^3 (t^5 −1))) = DNE

$$\mathrm{x}\:=\:\mathrm{t}^{\mathrm{60}} \\ $$$$\underset{\mathrm{t}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{t}^{\mathrm{20}} −\mathrm{t}^{\mathrm{12}} }{\mathrm{t}^{\mathrm{20}} −\mathrm{t}^{\mathrm{15}} }\:=\:\underset{\mathrm{t}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{t}^{\mathrm{12}} \left(\mathrm{t}^{\mathrm{8}} −\mathrm{1}\right)}{\mathrm{t}^{\mathrm{15}} \left(\mathrm{t}^{\mathrm{5}} −\mathrm{1}\right)} \\ $$$$=\:\underset{\mathrm{t}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{t}^{\mathrm{8}} −\mathrm{1}}{\mathrm{t}^{\mathrm{3}} \left(\mathrm{t}^{\mathrm{5}} −\mathrm{1}\right)}\:=\:\mathrm{DNE}\: \\ $$

Commented by Ar Brandon last updated on 07/May/20

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