Question

determine whether each limit is in the alternate f and the value, c, where the derivative is being eval
alternate form: $f(c)=lim_{x
ightarrow c}\frac{f(x)-f(c)}{x - c}$

1. $lim_{x

ightarrow4}\frac{sqrt{5x}-2sqrt{5}}{x - 4}$
f(x) = $sqrt{5x}$
c = 4

Explanation:

Step1: Identify the function and the value of c

Given $f(x)=\sqrt{5x}$ and $c = 4$. We use the alternate - form of the derivative $f^{\prime}(c)=\lim_{x
ightarrow c}\frac{f(x)-f(c)}{x - c}$.

Step2: Find the derivative of $y = f(x)=\sqrt{5x}=(5x)^{\frac{1}{2}}$

Using the power - rule $(u^n)^\prime=nu^{n - 1}u^\prime$, where $u = 5x$ and $n=\frac{1}{2}$. First, $u^\prime=5$. Then $y^\prime=f^\prime(x)=\frac{1}{2}(5x)^{-\frac{1}{2}}\times5=\frac{5}{2\sqrt{5x}}$.

Step3: Evaluate the derivative at $x = c$

Substitute $x = 4$ into $f^\prime(x)$. So $f^\prime(4)=\frac{5}{2\sqrt{5\times4}}=\frac{5}{2\times2\sqrt{5}}=\frac{\sqrt{5}}{4}$.

Answer:

$\frac{\sqrt{5}}{4}$