Evaluate the following limit: _ h 0 (a+h)^2 (a+h)-a^2 h - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{h}\rightarrow0}\frac{(\text{a}+\text{h})^2\sin(\text{a}+\text{h})-\text{a}^2\sin\text{a}}{\text{h}}$

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Answer

$\lim\limits_{\text{h}\rightarrow0}\frac{(\text{a}+\text{h})^2\sin(\text{a}+\text{h})-\text{a}^2\sin\text{a}}{\text{h}}$
$=\lim\limits_{\text{h}\rightarrow0}\frac{(\text{a}+\text{h})^2(\sin\text{a}\cos\text{h})-\text{a}^2\sin\text{a}+(\text{a}+\text{h})^2\cos\text{a}\sin\text{h}}{\text{h}}$
$=\lim\limits_{\text{h}\rightarrow0}\frac{\big(\text{a}^2+2\text{ah}+\text{h}\big)(\sin\text{a}\cos\text{h})-\text{a}^2\sin\text{a}+(\text{a}+\text{h})^2\cos\text{a}\sin\text{h}}{\text{h}}$
$=\lim\limits_{\text{h}\rightarrow0}\frac{\text{a}^2\sin\text{a}(\cos-1)+2\text{ah}\sin\text{a}\cos\text{h}+\text{h}^2\sin\text{a}\cos\text{h}+(\text{a}+\text{h})^2\cos\text{a}\sin\text{h}}{\text{h}}$
$=\lim\limits_{\text{h}\rightarrow0}\frac{\text{a}^2\sin\text{a}(\cos-1)}{\text{h}}+\lim\limits_{\text{h}\rightarrow0}\frac{2\text{ah}\sin\text{a}\cos\text{h}}{\text{h}}\\\ +\lim\limits_{\text{h}\rightarrow0}\frac{\text{h}^2\sin\text{a}\cos\text{h}}{\text{h}}+\lim\limits_{\text{h}\rightarrow0}\frac{(\text{a}+\text{h})^2\sin\text{a}\cos\text{h}}{\text{h}}$
$=\lim\limits_{\text{h}\rightarrow0}\frac{-\text{a}^2\sin\text{a}\sin^2\big(\frac{\text{h}}{2}\big)}{\frac{\text{h}}{2}}+2\text{a}\sin\text{a}+0+\text{a}^2\cos\text{a}$
$=0+2\text{a}\sin\text{a}+\text{a}^2\cos\text{a}$
$=2\text{a}\sin\text{a}+\text{a}^2\cos\text{a}$

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