Answer
$4x$
Work Step by Step
Step 1. Given $g(x)=2x^2+1$, we need to find the limit $g'(x)=\lim_{h\to0}\frac{g(x+h)-g(x)}{h}$
Step 2. We can find $g(x+h)=2(x+h)^2+1=2x^2+4xh+2h^2+1$ and $g(x+h)-g(x)=4xh+2h^2$
Step 3. Thus $g'(x)=\lim_{h\to0}\frac{g(x+h)-g(x)}{h}=\lim_{h\to0}\frac{4xh+2h^2}{h}=\lim_{h\to0}(4x+2h)=4x$
