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Problem 57 Hard Difficulty

If $ p $ is a polynomial, show that $ \displaystyle \lim_{x \to a}p(x) = p(a) $.


$a_{0}+a_{1} a+a_{2} a^{2}+\dots+a_{n} a^{n}=p(a)$


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Video Transcript

This is problem or fifty seven of the Stuart Calculus E It's edition section two point three. If p is a polynomial show that the limit is expert is a of this polynomial P of X is equal to p of a of the polynomial evaluated ex equally still, here we have generic panem eau palm a little here that we will try to solve using we basically middle laws and we're going to do is we're gonna play the first are one of the main limit loans has to do with tradition. So Lim is expert today of P won't be the same as element and six approaches a kn the first term, Russ No limit, as experts say of the next term, and so on. Our next step will be that Teo notice that the constants in front can be moved to the front. On this is a property kassidy with the concert hall limit law. Wait, man. Aksaray. The man is a person saying Teo, pull out the concept of this term a seven minus one Limited's expert His aim Thanks to that, my swan plus and so on. So those will continue on fairy tern here we have a limit of a function to a power for every term. And another way to write that using a ludmila is to take the limit of just this part the function razed to that same power we can also do this here is well in parentheses. Just the function X razed to that minutes one and so on. Plainly, What we have here now is that we have the exact same term, all of which will be the same as equal as a. We have a reduced one. This one big limited to Minnie's parliaments or the core of each of these limits is the limit is expert today of the function of X and since X is a linear function, a line that is continuous as your pro Jane clearly the limited C cool too, eh? And so what we have at the end is an get to the end and last one and then when this one and if we continue on, we should end with thanks a one time's a plus, some constant value that we started with the beginning and this is equal to the panem et evaluated at a and we have proven that the limit is expressive off. Have any polynomial as a function of X is equal to the apartment where you value with that, eh?