Science:Math Exam Resources/Courses/MATH110/December 2017/Question 02 (c)/Solution 1

By Hint 1, we recall that the derivative ${\textstyle f'(x)}$ of ${\textstyle f(x)}$ (at the point ${\textstyle x}$) is

$$f'(x)=\lim _{h\to 0}{\dfrac {f(x+h)-f(x)}{h}}.$$

At the point ${\textstyle x=0}$, the derivative is

$$f'(0)=\lim _{h\to 0}{\dfrac {f(h)-f(0)}{h}}.$$

Therefore, using the power rule, we have

$$\lim _{h\to 0}{\dfrac {f(h)-f(0)}{h}}=f'(0)=\left.100(x+1)^{99}\right|_{x=0}=100(0+1)^{99}=100.$$

Answer: ${\textstyle {\color {blue}{\mbox{True}}}}$.