Question

use limits to compute the derivative. f(1) where f(x)=x^{2}+4 f(1)= (simplify your answer.)

Explanation:

Step1: Recall the limit - definition of the derivative

The limit - definition of the derivative of a function $y = f(x)$ at $x = a$ is $f^{\prime}(a)=\lim_{h
ightarrow0}\frac{f(a + h)-f(a)}{h}$. Here, $a = 1$ and $f(x)=x^{2}+4$. First, find $f(1 + h)$ and $f(1)$.
$f(1+h)=(1 + h)^{2}+4=1 + 2h+h^{2}+4=h^{2}+2h + 5$.
$f(1)=1^{2}+4=5$.

Step2: Substitute into the limit - formula

$f^{\prime}(1)=\lim_{h
ightarrow0}\frac{f(1 + h)-f(1)}{h}=\lim_{h
ightarrow0}\frac{(h^{2}+2h + 5)-5}{h}$.
Simplify the numerator: $\frac{(h^{2}+2h + 5)-5}{h}=\frac{h^{2}+2h}{h}$.

Step3: Simplify the fraction

Since $h
eq0$ (as we are taking the limit as $h$ approaches 0, not setting $h = 0$), we can cancel out the $h$ in the numerator and denominator. $\frac{h^{2}+2h}{h}=h + 2$.

Step4: Evaluate the limit

Now, find $\lim_{h
ightarrow0}(h + 2)$. As $h$ approaches 0, we substitute $h = 0$ into $h + 2$. $\lim_{h
ightarrow0}(h + 2)=0+2=2$.

Answer:

$2$