Question

I need help with this problem. Please answer and explain! (Ignore selected answer)

Answer 1

Answer:

  g'(0) = 1

Step-by-step explanation:

The derivative of a function g at a number a, denoted by g'(a), is given by the definition of the derivative:

  $\displaystyle g'(a) = \lim_{h\to0} \dfrac{g(a+h) - g(a)}{h}$

In the definition of the derivative, "h" can be understood to be the horizontal change in the function with respect to a number a.

So g'(0) is

  $\displaystyle g'(0) = \lim_{h\to0} \dfrac{g(0+h) - g(0)}{h}=\lim_{h\to0} \dfrac{g(h) - g(0)}{h}$

The variable in the limit is bound; without any other variables around, we can change the variable name to another reasonable variable name and it will still be the same. Hence we can change h to x and it will be equivalent:

  $\displaystyle g'(0) = \lim_{x\to0} \dfrac{g(x) - g(0)}{x} = 1$

thus the given limit implies g'(0) = 1.