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 $f$.
-$( x ) = - 4 x + 7 , x = 6$
A) $- 4$
B) $- 24$
C) $- 17$
D) Does not exist

Question

Quizplus · Accepted Answer

Substituting $x=6$ and $f(x)=-4x+7$ into the limit expression, we get:
$$\lim_{h\rightarrow 0} \frac{f(6+h)-f(6)}{h} = \lim_{h\rightarrow 0} \frac{-4(6+h)+7 - (-4\cdot 6 + 7)}{h}$$
$$\qquad = \lim_{h\rightarrow 0} \frac{-4h}{h} = \lim_{h\rightarrow 0} -4 = -4$$
Therefore, the answer is choice A, $-4$.

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