Question

f(x)= $x^{3} + x^{2}$     e   determine $f(x+h) - f(x) / h$

Answer 1

f(x+h)=(x+h)³+(x+h)²=
(x³+3x²h+3xh²+h³)+(x²+2xh+h²)
f(x)=x³+x²
$\lim_{h \to 0 } (x+h)^{3} + (x+h)^{2} =$
$\lim_{h \to 0} \frac{( x^{3}+3 x^{2}h+3x h^{2}+ h^{3} )+( x^{2} +2xh+ h^{2})}{h}$
$\lim_{h \to 0} \frac{( x^{3} + x^{2} +3x^{2}h+3xh^{2}+2xh+h^{3}+h^{2}-x^{3}-x^{2})}{h}$
$\lim_{h \to 0} \frac{3x^{2}h+3xh^{2}+2xh+h^{3}+h^{2}}{h}$

$\lim_{n \to \infty} \frac{3x^{2}h}{h}+ \frac{3xh^{2}}{h}+ \frac{2xh}{h} + \frac{h^{3}}{h}+\frac{h^{2}}{h}$

$\lim_{h \to 0} 3x^{2}+3xh+2x+h^{2}+h$

substituindo h=0

$\lim_{h \to 0} 3x²+3x.0+2x+0²+0$

$\lim_{h \to 0}x^{3}+x^{2}=3x^{2}+2x$ #