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Question Number 162523 by cortano last updated on 30/Dec/21

   lim_(x→0)  ((7tan x−tan 7x)/x^3 ) =?

$$\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{7tan}\:{x}−\mathrm{tan}\:\mathrm{7}{x}}{{x}^{\mathrm{3}} }\:=? \\ $$

Answered by Ar Brandon last updated on 30/Dec/21

L=lim_(x→0) ((7tanx−tan7x)/x^3 )       =lim_(x→0) ((7(x+(x^3 /3))−(7x+(((7x)^3 )/3)))/x^3 )       =lim_(x→0) (1/x^3 )(((7x^3 )/3)−((343x^3 )/3))=((7−343)/3)=−112

$$\mathscr{L}=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{7tan}{x}−\mathrm{tan7}{x}}{{x}^{\mathrm{3}} } \\ $$$$\:\:\:\:\:=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{7}\left({x}+\frac{{x}^{\mathrm{3}} }{\mathrm{3}}\right)−\left(\mathrm{7}{x}+\frac{\left(\mathrm{7}{x}\right)^{\mathrm{3}} }{\mathrm{3}}\right)}{{x}^{\mathrm{3}} } \\ $$$$\:\:\:\:\:=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{1}}{{x}^{\mathrm{3}} }\left(\frac{\mathrm{7}{x}^{\mathrm{3}} }{\mathrm{3}}−\frac{\mathrm{343}{x}^{\mathrm{3}} }{\mathrm{3}}\right)=\frac{\mathrm{7}−\mathrm{343}}{\mathrm{3}}=−\mathrm{112} \\ $$

Answered by bobhans last updated on 30/Dec/21

 lim_(x→0)  ((7tan x−7x+7x−tan 7x)/x^3 ) =   lim_(x→0)  ((7(tan x−x))/x^3 ) −343 lim_(x→0)  ((tan 7x−7x)/((7x)^3 )) =   (7/3)−((343)/3) = −112

$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{7tan}\:\mathrm{x}−\mathrm{7x}+\mathrm{7x}−\mathrm{tan}\:\mathrm{7x}}{\mathrm{x}^{\mathrm{3}} }\:= \\ $$$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{7}\left(\mathrm{tan}\:\mathrm{x}−\mathrm{x}\right)}{\mathrm{x}^{\mathrm{3}} }\:−\mathrm{343}\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{tan}\:\mathrm{7x}−\mathrm{7x}}{\left(\mathrm{7x}\right)^{\mathrm{3}} }\:= \\ $$$$\:\frac{\mathrm{7}}{\mathrm{3}}−\frac{\mathrm{343}}{\mathrm{3}}\:=\:−\mathrm{112} \\ $$

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