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  • If <img alt="f(x)=\left\{\begin{matrix} \frac{\sin (a+2)x+\sin x}{x} ;&x<0 \\ b ;&x=0 \\ \frac{(x+3x^{2})^{\frac{1}{3}}-x^{\frac{1}{3}}}{x^{\frac{4}{3}}}; &x>0 \end{matrix}\r

If f(x)=\left\{\begin{matrix} \frac{\sin (a+2)x+\sin x}{x} ;&x<0 \\ b ;&x=0 \\ \frac{(x+3x^{2})^{\frac{1}{3}}-x^{\frac{1}{3}}}{x^{\frac{4}{3}}}; &x>0 \end{matrix}\right.  is continuous at x=0, then a+2b is equal to : 

 

Option: 1

0


Option: 2

1


Option: 3

-1


Option: 4

-2


Answers (1)

best_answer

 

 

Continuity -

A function f(x) is continuous at a point a if and only if the following three conditions are satisfied:

  1. f(a) is defined 

  2. \lim_{x\rightarrow a}f(x)\;exists

  3. \lim_{x\rightarrow a}f(x)=f(a)

Or\\\lim_{x\rightarrow a^-}f(x)=\lim_{x\rightarrow a^+}f(x)=f(a)\\\\i.e.\;\;\text{L.H.L. = R.H.L. value of a function at }x=a

-

L H L=\frac{\sin (a+2) x}{x}\frac{a+2}{a+2}+\frac{\sin x}{x}=a+3

R H L=\frac{(1+3 x)^{1 / 3}-1}{x}=1

\\a=-2,\;\;\;b=1\\a+2b=0

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vishal kumar

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