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Question
Evaluate `lim_(h -> 0) ((a + h)^2 sin (a + h) - a^2 sina)/h`
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Solution
We have `lim_(h -> 0) ((a + h)^2 sin (a + h) - a^2 sina)/h`
= `lim_(h -> 0) ((a^2 + h^2 + 2ah)[sina cos h + cos a sin h] - a^2 sin a)/h`
= `lim_(h -> 0) [(a^2 sin a (cos h - 1))/h + (a^2 + cos a sin h)/h + (h + 2a) (sin a cos h + cos a sin h)]`
= `lim_(h -> 0) (a^2 sin a (-2sin^2 h/2))/(h^2/2) * h/2 + lim_(h -> 0) (a^2 cos a sin h)/h + lim_(h -> 0) (h + 2a) sin(a + h)`
= `a^2 sin a xx 0 + a^2 cos a(1) + 2a sina`
= `a^2 cos a + 2a sina`.
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