Find the limit by evaluating the derivative of a suitable function at an appropriate value of x. (Hint: Use the definition of the derivative.) $\lim _{h \rightarrow 0} \frac{2(5+h)^{2}+5(5+h)-75}{h}$
A) 25
B) 75
C) 0
D) $\infty$

Question

Quizplus · Accepted Answer

Using the definition of the derivative, we have:
$$
\begin{aligned}
f'(x) &= \lim_{h 	o 0} \frac{f(x+h)-f(x)}{h} \
&= \lim_{h 	o 0} \frac{2(5+h)^2 + 5(5+h) - 75}{h} \
&= \lim_{h 	o 0} \frac{2(25+10h+h^2) + 25+5h - 75}{h} \
&= \lim_{h 	o 0} \frac{2h^2 + 20h}{h} \
&= \lim_{h 	o 0} (2h + 20) \
&= 20
\end{aligned}
$$
Therefore, $\lim_{h 	o 0} \frac{2(5+h)^2 + 5(5+h) - 75}{h} = f'(5) = 20$. So, the answer is A) 25.

Find the Limit by Evaluating the Derivative of a Suitable lim⁡h→02(5+h)2+5(5+h)−75h\lim _{h \rightarrow 0} \frac{2(5+h)^{2}+5(5+h)-75}{h}limh→0​h2(5+h)2+5(5+h)−75​

Find the Limit by Evaluating the Derivative of a Suitable $\lim _{h \rightarrow 0} \frac{2(5+h)^{2}+5(5+h)-75}{h}$