Maharashtra State BoardHSC Arts 12th Board Exam
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If the function f(x)=(5^sinx-1)^2/(xlog(1+2x))  for x ≠ 0 is continuous at x = 0, find f (0). - Mathematics and Statistics

If the function `f(x)=(5^sinx-1)^2/(xlog(1+2x))`  for x ≠ 0 is continuous at x = 0, find f (0).

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Solution

f is continuous at x = 0.

`f(0)=lim_(x->0)f(x)`

`f(0)=lim_(x->0)(5^sinx-1)^2/(xlog(1+2x))=lim_(x->0)((5^sinx-1)^2/x^2)/((xlog(1+2x))/x^2)`

`=lim_(x->0)(((5^sinx-1)/sinx)^2.sin^2x/x^2)/((2log(1+2x))/(2x))`

`=(lim_(x->0)(5^sinx-1)/sinx xx.lim_(x->0)sinx/x)^2/(2((lim_(x->0)log(1+2x))/(2x)))`

`f(0)=(log5)^2/2`

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