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 18587 by Joel577 last updated on 25/Jul/17

lim_(x→0) ((x . tan x)/(x sin x − cos x + 1))

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{x}\:.\:\mathrm{tan}\:{x}}{{x}\:\mathrm{sin}\:{x}\:−\:\mathrm{cos}\:{x}\:+\:\mathrm{1}} \\ $$

Commented by Joel577 last updated on 25/Jul/17

⇒ lim_(x→0) ((x . tan x)/(x sin x + 2sin^2 (1/2)x)) = lim_(x→0) ((x . tan x)/(sin x (x + 2sin (1/2)x))) ⇒ lim_(x→0) (x/(sin x)) . lim_(x→0) ((tan x)/(x + 2sin (1/2)x)) ⇒ 1 . lim_(x→0) (((tan x)/x)/((x + 2sin (1/2)x)/x)) = 1 . (1/(1 + 1)) = (1/2) Is this correct?

$$\Rightarrow\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{x}\:.\:\mathrm{tan}\:{x}}{{x}\:\mathrm{sin}\:{x}\:+\:\mathrm{2sin}^{\mathrm{2}} \:\frac{\mathrm{1}}{\mathrm{2}}{x}}\:=\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{x}\:.\:\mathrm{tan}\:{x}}{\mathrm{sin}\:{x}\:\left({x}\:+\:\mathrm{2sin}\:\frac{\mathrm{1}}{\mathrm{2}}{x}\right)} \\ $$$$\Rightarrow\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{x}}{\mathrm{sin}\:{x}}\:.\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{tan}\:{x}}{{x}\:+\:\mathrm{2sin}\:\frac{\mathrm{1}}{\mathrm{2}}{x}} \\ $$$$\Rightarrow\:\mathrm{1}\:.\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\frac{\mathrm{tan}\:{x}}{{x}}}{\frac{{x}\:+\:\mathrm{2sin}\:\frac{\mathrm{1}}{\mathrm{2}}{x}}{{x}}}\:=\:\mathrm{1}\:.\:\frac{\mathrm{1}}{\mathrm{1}\:+\:\mathrm{1}}\:=\:\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\mathrm{Is}\:\mathrm{this}\:\mathrm{correct}? \\ $$

Commented by mrW1 last updated on 26/Jul/17

no, you made a mistake. ⇒ lim_(x→0) ((x . tan x)/(x sin x + 2sin^2 (1/2)x)) ⇒ lim_(x→0) (((tan x)/x)/(((sin x)/x) +(1/2) (((sin (x/2))/(x/2)))^2 )) ⇒ (1/(1+(1/2)×1^2 ))=(2/3)

$$\mathrm{no},\:\mathrm{you}\:\mathrm{made}\:\mathrm{a}\:\mathrm{mistake}. \\ $$$$\Rightarrow\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{x}\:.\:\mathrm{tan}\:{x}}{{x}\:\mathrm{sin}\:{x}\:+\:\mathrm{2sin}^{\mathrm{2}} \:\frac{\mathrm{1}}{\mathrm{2}}{x}}\: \\ $$$$\Rightarrow\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\:\frac{\frac{\mathrm{tan}\:\mathrm{x}}{\mathrm{x}}}{\frac{\mathrm{sin}\:\mathrm{x}}{\mathrm{x}}\:+\frac{\mathrm{1}}{\mathrm{2}}\:\left(\frac{\mathrm{sin}\:\frac{\mathrm{x}}{\mathrm{2}}}{\frac{\mathrm{x}}{\mathrm{2}}}\right)^{\mathrm{2}} }\: \\ $$$$\Rightarrow\:\frac{\mathrm{1}}{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}×\mathrm{1}^{\mathrm{2}} }=\frac{\mathrm{2}}{\mathrm{3}} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com