Single Variable Calculus, Chapter 2, 2.4, Section 2.4, Problem 10

Find a number $\delta$ by using a graph such that if $5 < x < 5 + \delta \quad$ then $\displaystyle \quad \frac{x^2}{\sqrt{x-5}} > 100$


We can determine the value of $\delta$ by getting the point of intersection of the curve and the line $y=100$. As shown in the graph:






$\begin{equation} \begin{aligned} \frac{x^2}{\sqrt{x-5}} & = 100\\ x^2 &= 100\sqrt{x-5}\\ x^2 - 100 \sqrt{x-5} & = 0\\ \end{aligned} \end{equation}$

Solving for $x$,
$\quad x = 5.0659$


Hence, the value of $\delta$ is...


$\quad \begin{equation} \begin{aligned} 5 + \delta & \leq x \\ \delta & \leq x - 5\\ \delta & \leq 5.0659 -5\\ \delta & \leq 0.0659 \end{aligned} \end{equation}$