lim(y tends to 0) (a+h)²sin(a+h) - a²sina / h

solve the limit without using l- hospital rule

Although you want answer without L'hospital rule but if you are asked in the examination then you have to use the property that you have learnt in your book, so its good to use formula and here we are providing you the solution with using L'hospital rule, 
$\underset{h \to 0}{\lim } \frac{{\left(a + h\right)}^2 \sin (a + h) - a^2 \sin a}{h}$

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

$$\begin{gathered} \underset{h\to 0}{\lim }\frac{2(a+h)\sin (a+h)+(a+h{)}^2\cos (a+h)-0}{1} \\ = 2(a+0)\sin (a+0)+(a+0{)}^{ 2}\cos (a+0) \\ = 2a \sin a + a^2 \cos a \end{gathered}$$