To find limit of limx→0cos(sinx)−cosxx4.
I differentiated it using L Hospital's rule. I got
−sin(sinx)cosx+sinx4x3.
I divided and multiplied by sinx.
Since limx→0sinxx=1, thus I got
1−cosx4x2.On applying standard limits, I get answer 18. But correct answer is 16. Please help.
Answer
Using Prosthaphaeresis Formulas,
cos(sinx)−cosx=2sinx−sinx2sinx+sinx2
So, cos(sinx)−cosxx4=2sinx−sinx2x−sinx2sinx+sinx2x+sinx2⋅x−sinxx3⋅x+sinxx⋅14
We know, limh→0sinhh=1
Apply L'Hospital's Rule on limx→0x−sinxx3
and we get limx→0x+sinxx=1+limx→0sinxx
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