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  •  If<img alt="\mathrm{ \lim _{x \rightarrow 0}\left(\frac{\sin 2 x}{x^3}+a+\frac{b}{x^2}\right)=0}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B%20%5Clim%20_%7Bx%20%5Cright

 If\mathrm{ \lim _{x \rightarrow 0}\left(\frac{\sin 2 x}{x^3}+a+\frac{b}{x^2}\right)=0} then the value of \mathrm{ 3 a +b} is

Option: 1

2


Option: 2

-2


Option: 3

-1


Option: 4

0


Answers (1)

best_answer


\begin{aligned} &\mathrm{ \text { Required limit }=\lim _{x \rightarrow 0} \frac{\sin 2 x+a x^3+b x}{x^3}\left(\frac{0}{0}\right) }\\ &\mathrm {=\lim _{x \rightarrow 0} \frac{2 \cos 2 x+3 a x^2+b}{3 x^2} }\\ \end{aligned}

If this limit exists then, we must have b+2=0 In this case the last limit is equal to

\mathrm{\lim _{x \rightarrow 0} \frac{-4 \sin 2 x+6 a x}{6 x^2}=-\frac{4}{3}+a \text {, so } \mathrm{a}=\frac{4}{3} \text {. } }

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vinayak

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