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Question Number 107163 by bobhans last updated on 09/Aug/20

Answered by john santu last updated on 09/Aug/20

⊵JS⊴ lim_(x→0) cos x . lim_(x→0) ((x(√(1+(5/x))))/(2(((1−((sin x)/8)))^(1/5) −1)))= 1 × lim_(x→0) ((x(√(1+(5/x))))/(2(1−((sin x)/(40))−1)))= −20×lim_(x→0) ((x(√(1+(5/x))))/(sin x)) = −∞

$$\:\:\:\trianglerighteq\mathrm{JS}\trianglelefteq \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\mathrm{cos}\:\mathrm{x}\:.\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{x}\sqrt{\mathrm{1}+\frac{\mathrm{5}}{\mathrm{x}}}}{\mathrm{2}\left(\sqrt[{\mathrm{5}}]{\mathrm{1}−\frac{\mathrm{sin}\:\mathrm{x}}{\mathrm{8}}}−\mathrm{1}\right)}= \\ $$$$\mathrm{1}\:×\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{x}\sqrt{\mathrm{1}+\frac{\mathrm{5}}{\mathrm{x}}}}{\mathrm{2}\left(\mathrm{1}−\frac{\mathrm{sin}\:\mathrm{x}}{\mathrm{40}}−\mathrm{1}\right)}= \\ $$$$−\mathrm{20}×\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{x}\sqrt{\mathrm{1}+\frac{\mathrm{5}}{\mathrm{x}}}}{\mathrm{sin}\:\mathrm{x}}\:=\:−\infty\: \\ $$

Commented by bobhans last updated on 09/Aug/20

thank you

$$\mathrm{thank}\:\mathrm{you} \\ $$

Answered by Dwaipayan Shikari last updated on 09/Aug/20

lim_(x→0) ((cosx(√(x^2 +5x)))/(((32−4sinx))^(1/5) −2)).((32−4sinx−32)/(−4sinx)) lim_(x→0) ((√(x^2 +5x))/(−4sinx)).5.2^4 →−∞

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{cosx}\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{5x}}}{\sqrt[{\mathrm{5}}]{\mathrm{32}−\mathrm{4sinx}}\:−\mathrm{2}}.\frac{\mathrm{32}−\mathrm{4sinx}−\mathrm{32}}{−\mathrm{4sinx}} \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{5x}}}{−\mathrm{4sinx}}.\mathrm{5}.\mathrm{2}^{\mathrm{4}} \rightarrow−\infty \\ $$

Commented by john santu last updated on 09/Aug/20

i don′t understand your process lim_(x→0) ((cos x (√(x^2 +5x)))/(((32−4sin x))^(1/5) −2)) . ((32−4sin x−32)/(−4sin x)) ??? what formula it is?

$$\mathrm{i}\:\mathrm{don}'\mathrm{t}\:\mathrm{understand}\:\mathrm{your}\:\mathrm{process} \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{cos}\:\mathrm{x}\:\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{5x}}}{\sqrt[{\mathrm{5}}]{\mathrm{32}−\mathrm{4sin}\:\mathrm{x}}−\mathrm{2}}\:.\:\frac{\mathrm{32}−\mathrm{4sin}\:\mathrm{x}−\mathrm{32}}{−\mathrm{4sin}\:\mathrm{x}} \\ $$$$???\:\mathrm{what}\:\mathrm{formula}\:\mathrm{it}\:\mathrm{is}?\: \\ $$

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