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Question

limx0sin(a+x)+sin(ax)2sinaxsinx

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Solution

We have,

limx0sin(a+x)sin(ax)2sinaxsinx

=limx02sin(a+x+ax)2cos(a+xa+x)22sinaxsinx

=limx02sin2a2cos(2x)22sinaxsinx

=limx02sinacosx2sinaxsinx

=limx02sina(cosx1)xsinx

=limx02sina(12sin2x21)xsinx

=limx04sinasin2x2xsinx

=limx04sinasin2x2x2sinx2cosx2

=limx02sinasinx2xcosx200form

Applying L’ Hospital rule and we get,

=limx02sinacosx2×(12)x(sinx2)12+cosx2

=limx0sinacosx2x2(sinx2)+cosx2

Taking limit and we get,

=sinacos00(sin0)+cos0

=sina0+1

=sina

Hence, this is the answer.

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