Question and Answers Forum

All Questions      Topic List

Limits Questions

Previous in All Question      Next in All Question      

Previous in Limits      Next in Limits      

Question Number 138153 by mathlove last updated on 10/Apr/21

lim_(x→0^+ ) ((x−∣tanx∣)/(∣sinx∣−x))=?

$$\underset{{x}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\frac{{x}−\mid{tanx}\mid}{\mid{sinx}\mid−{x}}=? \\ $$

Answered by TheSupreme last updated on 10/Apr/21

for x→0^+  ∣sin(x)∣=sin(x) and ∣tan(x)∣=tan(x)  lim ((x−tan(x))/(sin(x)−x))=  for x→0  tan(x)→x+(x^3 /3)  sin(x)→x−(x^3 /6)  lim.... = lim ((−(x^3 /3))/(−(x^3 /6)))=2

$${for}\:{x}\rightarrow\mathrm{0}^{+} \:\mid{sin}\left({x}\right)\mid={sin}\left({x}\right)\:{and}\:\mid{tan}\left({x}\right)\mid={tan}\left({x}\right) \\ $$$${lim}\:\frac{{x}−{tan}\left({x}\right)}{{sin}\left({x}\right)−{x}}= \\ $$$${for}\:{x}\rightarrow\mathrm{0} \\ $$$${tan}\left({x}\right)\rightarrow{x}+\frac{{x}^{\mathrm{3}} }{\mathrm{3}} \\ $$$${sin}\left({x}\right)\rightarrow{x}−\frac{{x}^{\mathrm{3}} }{\mathrm{6}} \\ $$$${lim}....\:=\:{lim}\:\frac{−\frac{{x}^{\mathrm{3}} }{\mathrm{3}}}{−\frac{{x}^{\mathrm{3}} }{\mathrm{6}}}=\mathrm{2} \\ $$

Commented by greg_ed last updated on 10/Apr/21

cool !  i′ve tried for x→0^−  and the answer is −1. :)

$$\mathrm{cool}\:! \\ $$$$\left.\mathrm{i}'\boldsymbol{\mathrm{ve}}\:\boldsymbol{\mathrm{tried}}\:\boldsymbol{\mathrm{for}}\:\boldsymbol{{x}}\rightarrow\mathrm{0}^{−} \:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{answer}}\:\boldsymbol{\mathrm{is}}\:−\mathrm{1}.\::\right) \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com