Friday, 19 September 2014

Calculating limit of function



To find limit of limx0cos(sinx)cosxx4.
I differentiated it using L Hospital's rule. I got
sin(sinx)cosx+sinx4x3.

I divided and multiplied by sinx.

Since limx0sinxx=1, thus I got
1cosx4x2.On applying standard limits, I get answer 18. But correct answer is 16. Please help.


Answer



Using Prosthaphaeresis Formulas,



cos(sinx)cosx=2sinxsinx2sinx+sinx2



So, cos(sinx)cosxx4=2sinxsinx2xsinx2sinx+sinx2x+sinx2xsinxx3x+sinxx14



We know, limh0sinhh=1




Apply L'Hospital's Rule on limx0xsinxx3



and we get limx0x+sinxx=1+limx0sinxx


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