Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ occur frequently in calculus. Evaluate this limit for the given value of $x$ and function $f$.
-$$( x ) = 5 x ^ { 2 } + 4 , x = - 4$$
A) Does not exist
B) 80
C) $- 36$
D) $- 40$

Question

Quizplus · Accepted Answer

We plug in the given value of $x$ and the function into the limit expression:
\begin{align*} \lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } &= \lim _ { h \rightarrow 0 } \frac { f ( -4 + h ) - f ( -4 ) } { h } \ &= \lim _ { h \rightarrow 0 } \frac { 5 ( -4 + h ) ^ { 2 } + 4 - 84 } { h } \ &= \lim _ { h \rightarrow 0 } \frac { 5 ( 16 - 8 h + h ^ { 2 } ) - 80 } { h } \ &= \lim _ { h \rightarrow 0 } \frac { 80 - 40 h + 5 h ^ { 2 } - 80 } { h } \ &= \lim _ { h \rightarrow 0 } ( -40 + 5 h ) \ &= -40 \end{align*}
Therefore, the answer is D.

Because of Their Connection with Secant Lines, Tangents, and Instantaneous 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)​

Because of Their Connection with Secant Lines, Tangents, and Instantaneous $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$