Question

question given the function f(x) = √x, which of the following is the correct limit definition of f(4)? select the correct answer below: lim h→0 (√(4 + h)/h) lim h→0 ((√(4 + h) - 2)/h) lim h→0 ((2 - √h)/h) lim h→0 ((√(4 - h) - √h)/h)

Explanation:

Step1: Recall derivative limit - definition

The limit - definition of the derivative of a function $y = f(x)$ at $x = a$ is $f^{\prime}(a)=\lim_{h
ightarrow0}\frac{f(a + h)-f(a)}{h}$.

Step2: Identify $f(x)$, $a$ values

Given $f(x)=\sqrt{x}$ and $a = 4$. Then $f(4)=\sqrt{4}=2$ and $f(4 + h)=\sqrt{4 + h}$.

Step3: Substitute into the formula

Substitute $f(4 + h)$ and $f(4)$ into the derivative limit - definition: $f^{\prime}(4)=\lim_{h
ightarrow0}\frac{f(4 + h)-f(4)}{h}=\lim_{h
ightarrow0}\frac{\sqrt{4 + h}-2}{h}$.

Answer:

$\lim_{h
ightarrow0}\frac{\sqrt{4 + h}-2}{h}$ (the second option)