Question

Given the function g(x)= 5-x^2, simplify the bottom

Answer 1

Answer:

$-2x$

Step-by-step explanation:

We are given the function $g(x)=5-x^2$ and the expression$\lim_{x \to 0}  \frac{g(x+h)-g(x)}{h}$

First, we must substitute in x+h into g(x) and substitute in g(x) into the expression

$\lim_{h \to 0}\frac{5-(x+h)^2-(5-x^2)}{h}$

Next, we can simplify by multiplying out the exponents and then combine like terms

$\lim_{h \to 0}\frac{5-(x+h)^2-(5-x^2)}{h} \\\\\lim_{h \to 0}\frac{5-x^2-2xh-h^2-5+x^2}{h} \\\\\lim_{h \to 0}\frac{-2xh-h^2}{h}$

Next, we can divide the numerator by the denominator to get

$\lim_{h \to 0}\frac{-2xh-h^2}{h}\\\\\lim_{h \to 0} -2x-h$

Lastly, we can substitute in 0 for h

$-2x-0\\\\-2x$