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  • If <img alt="\mathrm{f(x)=\left\{\begin{array}{ll}a+\frac{\sin [x]}{x}, & x>0 \\ 2, & x=0 \\ b+\left[\frac{\sin x-x}{x^3}\right], & x<0\end{array}\right.,}" src="https://entrancec

If \mathrm{f(x)=\left\{\begin{array}{ll}a+\frac{\sin [x]}{x}, & x>0 \\ 2, & x=0 \\ b+\left[\frac{\sin x-x}{x^3}\right], & x<0\end{array}\right.,} (where [.] denotes the greatest integer function). If f(x) is continuous at \mathrm{x=0,} then b is equal to

Option: 1

\mathrm{a-1}


Option: 2

\mathrm{a+1}


Option: 3

\mathrm{a+2}


Option: 4

\mathrm{a-2}


Answers (1)

best_answer

\mathrm {\begin{aligned} & f(0+0)=a \text { as } \sin [x]=0 \text { as } x \rightarrow 0+0 \\ & f(0-0)=b-1 \text { as } \frac{\sin x-x}{x^3}=-\frac{1}{6}+\frac{x}{4 !}-\frac{x^2}{5 !}+\ldots \ldots . . \\ & f(0)=2 \\ & \Rightarrow a=2 ; b=3 \end{aligned}}

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