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Numerically and graphically determine a $\delta$ corresponding to (a) $\epsilon=\boldsymbol{0} .1$ and $(\mathbf{b}) \varepsilon=\mathbf{0 . 0 5 .}$ Graph the function in the $\epsilon-\delta$ window $[x \text { -range is }(a-\delta, a+\delta) \text { and } y$ -range is $(L-\varepsilon, L+\varepsilon)]$ to verify that your choice works.$$\lim _{x \rightarrow 0}\left(x^{2}+1\right)=1$$

a. $\delta=\sqrt{0.1}$b. $\left|\left(x^{2}+1\right)-1\right|<0.1$

Calculus 1 / AB

Chapter 1

Limits and Continuity

Section 6

Formal Definition of the Limit

Limits

Continuous Functions

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Okay, so here we are concerned the limit as X approaches zero of X squared plus one is equal to one. And the objective here is to numerically and graphically determine a delta corresponding to an epsilon that's equal to 0.1. Okay, so the precise definition off limit states that well, for a function in some open interval, we have the limit as X approaches A of our function after Becks. Okay, that is equal G L. So if well and only if, um for any absolute reason, Joe. So for all Absalon greater than zero, we have a daughter. So there exists in a Delta Graydon zero. What's that? Well, such that if zero is less than the absolute value of X minus day, yeah, value of X minus A, which is less than delta. Okay, that I'm implies back guarantees that the absolute value of the function absolute value of halfbacks minus your limit is less than absolute. Okay, um, so the go ahead and determine our delta here. Well, we have. So we'll be, perhaps that we start with the absolute value of FX minus out and well, this is equal to our function. Is X squared plus one. So we have absolute value of after tax is the absolute value of X squared plus one minus our limit. So minus one. Okay. What? That less than Epsilon, which were given UPS line of 0.1. Okay, so this is equivalent to saying Well, negative 0.1 is less than X squared, which is less than 0.1. Okay. And this is again equivalent to saying X squared is less than 0.1, which would you say, is quite readable. Science. And we get that ax or the absolute value of acts is going to be less than square root. It was 0.1 I'm rounds to 0.32 was less than 0.32 Okay, so therefore the absolute value of well X minus zero, we could say is less than 0.32 Therefore, our doctor hero is going to be equal to 0.32 Okay, So looking bad at and graphically. Okay, So here on Desmond was so here We have our function graph here. This is X squared plus one, and we look at our well, you know Absalon, Doctor window with the X range here goes from zero minus 3.32 There's native 0.32 to 0 plus 3.3 to start out the window and that are y range goes from, um, why minus 0.1 two y plus 0.1 on this primitive verifies the answer here we see that you know, for any Delta, right? If in the doctor window if X minus A if you have to bribe packs might say is between zero and Delta. That implies that ever backs minus, um minus limits here is going to be less than absolute. Okay, um, so then, looking at part B were given a different doctor. So here is our cult of the party 7.3 to then for, um for be Well, um were given an epsilon of 0.5 Now are absalon for part B is absolutely equal. Choose 0.5 sorted out. They don't to hear, um, pretty much the same exact way. We say that when we start with the end, we start with the absolute value of after backs minus l being less than absolute in. Okay, that means that the absolute value of after backs, which is our function X squared plus one minus the limit. L is going to be less than in this case, 0.5 So this is equipment the same. Negative 0.5 is less than X squared. Um, well, plus one minus one. So which is less than, um, 0.5? Okay. And this is again equipment to sing at the absolute value of X squared, um is less than 0.5 So the absolute value right of X squared is bus, Then 0.5 students quit. Read about science. We get that the absolute value of acts is going to be less than square root of 0.5 which is gonna be zero points by t two. So this is less than 0.22 So therefore, our delta here, um, is going to be equal to 0.22 and state we're in your head and draw the graph, um, for function after Rex being equal to X squared susswein in the absolute doctor window, where the X rains now goes from zero minus zero point to to 20 Plus, is there a 0.22? And the UAE range is on X range and are why? Range goes from? Well, one minus 0.5 to 1 plus 0.5 And we could verify that in the same way. We just changed our X value here. Um, no exes going from Make it a point to you Gotta do the point to case we see a new doctor range here and we see that we work in the doctor range. Prayed for all, Um so if for any or any acts right in this in the delta window, basically here guarantees that ffx minus el is gonna be less than the Given Absalon. Now. So our new palm are delta here corresponds to their appointed

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