If $ f(x)=-2 x^{2}+x $, find the following limit, if it exists.
$\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$
A) $ -4 x+1 $
B) $ -2 x+1 $
C) $ 2 x+1 $
D) $ -4 x $
E) The limit does not exist.

Question

Quizplus · Accepted Answer

The limit represents the derivative of $f(x)$ at $x$, which can be found using the definition of the derivative. Calculating $f(x+h)$ gives $-2(x+h)^2+(x+h)$, and subtracting $f(x)$ from this gives $-2(x^2+2xh+h^2)+x+h+2x^2-x$. Simplifying this and dividing by $h$ gives $-4x-2h+1$. Taking the limit as $h$ approaches 0 gives $-4x+1$.

If f(x)=−2x2+x f(x)=-2 x^{2}+x f(x)=−2x2+x , Find the Following Limit, If It Exists

If $f(x)=-2 x^{2}+x$ , Find the Following Limit, If It Exists