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 77182 by jagoll last updated on 04/Jan/20

    evaluate lim_(x→0)  ((∫_a ^x (((cos t)/t))dt)/x) .

$$ \\ $$$$ \\ $$$$\mathrm{evaluate}\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\underset{\mathrm{a}} {\overset{\mathrm{x}} {\int}}\left(\frac{\mathrm{cos}\:\mathrm{t}}{\mathrm{t}}\right)\mathrm{dt}}{\mathrm{x}}\:. \\ $$

Commented by mathmax by abdo last updated on 05/Jan/20

i think that a=0...

$${i}\:{think}\:{that}\:{a}=\mathrm{0}... \\ $$

Answered by john santu last updated on 04/Jan/20

suppose F(t) is antiderivative  ∫ ((cos t)/t) dt such that ((dF(t))/dt)= ((cos (t))/t).  now lim_(x→0)  ((F(t)∣_a ^x )/x)= lim_(x→0)  ((F(x)−F(a))/x)  we assume that F(0)−F(a)=0  lim_(x→0)  ((F′(x)−F′(a))/1)= F′(0)−F′(a)  but F′(0) = ((cos (0))/0) undefined  i think it problem error.

$${suppose}\:{F}\left({t}\right)\:{is}\:{antiderivative} \\ $$$$\int\:\frac{\mathrm{cos}\:{t}}{{t}}\:{dt}\:{such}\:{that}\:\frac{{dF}\left({t}\right)}{{dt}}=\:\frac{\mathrm{cos}\:\left({t}\right)}{{t}}. \\ $$$${now}\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{F}\left({t}\right)\underset{{a}} {\overset{{x}} {\mid}}}{{x}}=\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{F}\left({x}\right)−{F}\left({a}\right)}{{x}} \\ $$$${we}\:{assume}\:{that}\:{F}\left(\mathrm{0}\right)−{F}\left({a}\right)=\mathrm{0} \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{F}'\left({x}\right)−{F}'\left({a}\right)}{\mathrm{1}}=\:{F}'\left(\mathrm{0}\right)−{F}'\left({a}\right) \\ $$$${but}\:{F}'\left(\mathrm{0}\right)\:=\:\frac{\mathrm{cos}\:\left(\mathrm{0}\right)}{\mathrm{0}}\:{undefined} \\ $$$${i}\:{think}\:{it}\:{problem}\:{error}. \\ $$$$ \\ $$

Commented by jagoll last updated on 04/Jan/20

thanks sir. i will check my question

$$\mathrm{thanks}\:\mathrm{sir}.\:\mathrm{i}\:\mathrm{will}\:\mathrm{check}\:\mathrm{my}\:\mathrm{question} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com