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Use the definition of continuity and the properties of limits to show that the function is continuous at the given number $ a $.

$ p(v) = 2 \sqrt{3v^2 + 1}, \hspace{5mm} a = 1 $

\[\begin{aligned}\lim _{\boldsymbol{v} \rightarrow 1} p(v) &=\lim _{v \rightarrow 1} 2 \sqrt{3 v^{2}+1}=2 \lim _{v \rightarrow 1} \sqrt{3 v^{2}+1}=2 \sqrt{\lim _{v \rightarrow 1}\left(3 v^{2}+1\right)}=2 \sqrt{3 \lim _{v \rightarrow 1} v^{2}+\lim _{v \rightarrow 1} 1} \\&=2 \sqrt{3(1)^{2}+1}=2 \sqrt{4}=4=p(1)\end{aligned}\]By the definition of continuity, $p$ is continuous at $a=1$

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 5

Continuity

Limits

Derivatives

Campbell University

University of Michigan - Ann Arbor

University of Nottingham

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This is problem number thirteen of this to a calculus safe addition. Section two point five use the definition of continuity and the properties of limits to show that the function is continuous at the given number, eh, P A V is equal to two times a function square root over three B squared plus one at equals to one. And the definition of continuity in this case is that element on screen perches, eh? Of the function Pee Evy, this must be called a P urban being. Okay, so we're gonna I use our limit laws to figure out what the limit will be. Ah, our first limit along is Thie Consulate Miller that allows us to move out any constant to the front, over the limit. And he is approaching one, which is what it is. And it is this function that's left over square root of three B squared plus one. Our next step is to work on things that the square root is Ah, the same as a power of one F. And her power line says that we can all right, take the limit of the function on the inside and reason to that same power. So I have limited of dysfunction. Three piece cripples one, and this limit is going to be now raised to the one after hour and then inside. What we have is we have a sum of two terms. And so we needed to someone in combination with the constant room to simplify this one limited to two more simple women, three times the limit of severe purchase. One of these squared plus the limit and your purchase one one. Finally, we'LL be able to evaluate these easier limits, and we should be able to reduce it down to two times of twenty three Time's limited to be a purchase. One of B squared is equal to one squared eliminates. We have purchased one of one that's equal to one. And then here, we see is we get four on the inside. The school before is equal to four or two two times two Outside, too was left over is for, and that is what our limit is equal to. If we were to evaluate TV directly, kind of plugging in and they get one, you're positive one into our function, uh, would actually give us the same result two Times Square root of the quantity three times one squared plus one, which is the same as two times screwed before, Which is two temps, too. Four. And since this limited calls the from Jenny value what it may we see that this function is continuous, Addie.

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