Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ for $f ( x ) = 4 - 6 x$.
A) 6
B) 10
C) $- 6$
D) 4
E) limit does not exist

Question

Quizplus · Accepted Answer

This limit represents the definition of the derivative of $f(x)$ at $x$. To find it, substitute $f(x + h) = 4 - 6(x + h)$ and $f(x) = 4 - 6x$ into the limit expression. Simplifying gives $\lim _ { h \rightarrow 0 } \frac { 4 - 6(x + h) - (4 - 6x) } { h } = \lim _ { h \rightarrow 0 } \frac { -6h } { h } = -6$, which means the derivative of $f(x) = 4 - 6x$ is $-6$.

Find lim⁡h→0f(x+h)−f(x)h\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }limh→0​hf(x+h)−f(x)​

Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$