If If $f ( x ) = - 3 x ^ { 2 } - 4 x$, find the following limit, if it exists.
$$\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$$
A) $- 6 x - 4$
B) $- 3 x - 4$
C) $2 x - 4$
D) $- 6 x$
E) The limit does not exist.

Question

Quizplus · Accepted Answer

The limit represents the derivative of $f(x)$ at $x$, which is found using the definition of the derivative. Substituting $f(x + h)$ and $f(x)$ into the limit gives $\lim _ { h \rightarrow 0 } \frac { -3(x+h)^2 - 4(x+h) + 3x^2 + 4x } { h }$. Simplifying this expression and taking the limit as $h$ approaches 0 yields $-6x - 4$, which is the derivative of $f(x)$ with respect to $x$.

If If f(x)=−3x2−4xf ( x ) = - 3 x ^ { 2 } - 4 xf(x)=−3x2−4x

If If $f ( x ) = - 3 x ^ { 2 } - 4 x$