Answer
$f'(x)=2x+1$
Work Step by Step
To take the derivative of a function using the limit process, plug into the equation $f'(x)=\lim\limits_{h \to 0}\frac{f(x+h)-f(x)}{h}$ and simplify:
$f'(x)=\lim\limits_{h \to 0}\frac{[(x+h)^{2}+(x+h)-3]-(x^2+x-3)}{h}$
$f'(x)=\lim\limits_{h \to 0}\frac{x^2+2xh+h^2+x+h-3-x^2-x+3}{h}$
$f'(x)=\lim\limits_{h \to 0}\frac{2xh+h^2+h}{h}$
$f'(x)=\lim\limits_{h \to 0}2x+h+1$
Once you can't simplify any further, plug 0 in for $h$:
$f'(x)=2x+(0)+1=2x+1$
