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### HOW DO YOU SEE IT? Use the graph of $f$ to identi…

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University of Rochester
Problem 71

# Graphical Reasoning The statement$$\lim _{x \rightarrow 2} \frac{x^{2}-4}{x-2}=4$$means that for each $\varepsilon>0$ there corresponds a $\delta>0$ such that if $0<|x-2|<\delta,$ then$$\left|\frac{x^{2}-4}{x-2}-4\right|<\varepsilon$$If $\varepsilon=0.001,$ then$$\left|\frac{x^{2}-4}{x-2}-4\right|<0.001$$Use a graphing utility to graph each side of this inequality. Use the zoom feature to find an interval $(2-\delta, 2+\delta)$ such that the inequality is true.

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## Video Transcript

so we're gonna graph the equation. The inequality X square plus Y minus two square is less than or equal to four. Well, first of all, we can identify these a circle because it has a four X square was why square is equal to our square. So right away, we can say this is a circle with a radius too. So are equals two with a center at zero come out to. So this circle would Ray just do that has shifted to units. Its centre has shifted to units from the origin so we can take the point circle zero. So our center Sorry the center is gonna be There's one that's here. Oh, come on. Two has a radius of two. So it goes all the way to zero. It goes all the way right here just and it looks something like this. It's just an approximation. So this is the equation. The graph of the equation. Why is equal? Sorry. Um, this is a graph of equation X square plus why minus d'oh square equals four. As it's a graph se inequality. We will need to take to test points one inside the circle. I'm just gonna take the center, which is just gonna be the point zero. Come on. Two, one outside of it. I'm just gonna take the point zero comma minus. And remember, you can take any point you want any two points that you want one inside of the circle and one outside. Just to make sure which region we're gonna be shading, We're gonna sit substantive, been back into the inequality. So, for when zero commit to zero comma, too, we will be having zero square, which is a value fax plus to minus two square. It's last an oracle to four, and we know this. Both quantities are just zero. So both of them one of these are just zero. So we have zero supplies ventricle before. So this checks for our inequality for an hour. Another point zero comma minus two. So we will be having zero square again plus minus two minus two square Lays an oracle to four. So we know minus two minus Jews, maybe four. So we have made it for squares. Last an oracle to four. So negative four square is just 16 16 last miracles for and we know this is true. This is not satisfied than quality, therefore are shaded. Region will be the one inside the circle and we have a less than or equal to four were including all of the values on the lying on the equation X square plus why minus juice square is equal to four. So all of this is represented by the solid line. I'm closing the circle at the shaded region. It's gonna be inside the circle.

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