Question

find lim h→0 (f(5 + h) - f(5))/h if f(x)= - 7x - 1. lim h→0 (f(5 + h) - f(5))/h = □ (simplify your answer.)

Explanation:

Step1: Find f(5 + h)

Substitute x = 5 + h into f(x)=-7x - 1.
$f(5 + h)=-7(5 + h)-1=-35-7h - 1=-36-7h$

Step2: Find f(5)

Substitute x = 5 into f(x)=-7x - 1.
$f(5)=-7\times5-1=-35 - 1=-36$

Step3: Substitute into the limit expression

$\lim_{h
ightarrow0}\frac{f(5 + h)-f(5)}{h}=\lim_{h
ightarrow0}\frac{(-36-7h)-(-36)}{h}$
$=\lim_{h
ightarrow0}\frac{-36-7h + 36}{h}=\lim_{h
ightarrow0}\frac{-7h}{h}$

Step4: Simplify the limit

Cancel out h in the numerator and denominator.
$\lim_{h
ightarrow0}\frac{-7h}{h}=\lim_{h
ightarrow0}-7=-7$

Answer:

-7