lim(y tends to 0) (a+h)^{2}sin(a+h) - a^{2}sina / h

solve the limit without using l- hospital rule

$\underset{h\to 0}{\mathrm{lim}}\frac{{\left(a+h\right)}^{2}\mathrm{sin}(a+h)-{a}^{2}\mathrm{sin}a}{h}$

this is in $\frac{0}{0}$ form, so we applying L- hospital's rule, so we differentiate the fraction,

$\underset{h\to 0}{\mathrm{lim}}\frac{2(a+h)\mathrm{sin}(a+h)+(a+h{)}^{2}\mathrm{cos}(a+h)-0}{1}\phantom{\rule{0ex}{0ex}}=2(a+0)\mathrm{sin}(a+0)+(a+0{)}^{2}\mathrm{cos}(a+0)\phantom{\rule{0ex}{0ex}}=2a\mathrm{sin}a+{a}^{2}\mathrm{cos}a$

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