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:
[tex]\displaystyle g'(a) = \lim_{h\to0} \dfrac{g(a+h) - g(a)}{h}[/tex]
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
[tex]\displaystyle g'(0) = \lim_{h\to0} \dfrac{g(0+h) - g(0)}{h}=\lim_{h\to0} \dfrac{g(h) - g(0)}{h}[/tex]
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:
[tex]\displaystyle g'(0) = \lim_{x\to0} \dfrac{g(x) - g(0)}{x} = 1[/tex]
thus the given limit implies g'(0) = 1.
