Question

if p is a polynomial how do i show that the limit of p(x) as x approaches a = p(a)

Answer 1

\[\lim_{x \rightarrow a}p(x)=p(a)\]

Answer 2

well... i need some more info first

Answer 3

what type of class is this for, analysis or calc?

Answer 4

calc

Answer 5

oook, uhm well, crap... hmm what arsenal do you have so far? Have you done the addition of limits?

Answer 6

yeah

Answer 7

ok so can you tell me what every single polynomial looks like?

Answer 8

the general form of a polynomial

Answer 9

x^2+x+c

Answer 10

well, almost what about a 5th deg poly? or a 10th degree?

Answer 11

x^n+x^n-1+...+c

Answer 12

how about coefficients too?

Answer 13

anx^n+an-1x^(n-1)+an-2x^(n-2)+...a0

Answer 14

good, ok, so now, what if you took the limit of that

Answer 15

could you split it up?

Answer 16

wouldnt you just get a's instead of x's

Answer 17

no, not quite, what happens is you change it into the sum of limits, which hopefully you have said \(lim_{x\rightarrow c} x^2=c^2\)

Answer 18

and similar, so you can just apply that then undo in the end

Answer 19

So we want \[\lim_{x\rightarrow c} a_nx^n+a_{n-1}x^{(n-1)}+a_{{n-2}}x^{(n-2)}+...+a_0=?\]

Answer 20

We can use addition there right to split up the limit?

Answer 21

yeah \[\lim_{x \rightarrow c}a _{n}x ^{n}+\lim_{x \rightarrow c} a _{n-1}x ^{(n-1)}+....\] like this right?

Answer 22

good, now, have you shown in class what \[lim_{x\rightarrow c} x^n=?\]

Answer 23

isn't it just \[c^n\]

Answer 24

it is, but if you haven't shown it, we can't use it :/

Answer 25

oh how do you show it

Answer 26

well, have you done product of limits? or have you learned the epsilon delta definition of a limit?

Answer 27

no

Answer 28

you haven't done product of a limit is the limit of the product?

Answer 29

i've done the product of limits but not epsilon delta proof

Answer 30

ok that's fine then, so now, how else can you write x^n?

Answer 31

x multiplied by n number of x's

Answer 32

good good, ok so limit of x?

Answer 33

as x goes to c, sorry forgot to mention

Answer 34

c?

Answer 35

yes, and I'm really hoping you've covered that or else based off of info given, you cannot do this problem

Answer 36

i guess you could just say it, it is a definition after all

Answer 37

oh ok, we weren't really shown in depth on the properties of limits and stuff, so i'm a bit lost

Answer 38

well, using \(\epsilon-\delta\) you can prove it but you just have to accept that

Answer 39

oh ok, so. Let's say the only things you know are that lim x=c and lim k=k if k is a constant

Answer 40

So, a limit can be distributed and factored but you must remember it is a function so it cannot be divided away

Answer 41

so you have to use a 2 step proof here. First, you need to show that lim ax^n=ac^n, then use that to show your big lim p(x)=p(c)

Answer 42

if you would like a better explanation of the limit properties you can look on khan academy and purplemath

Answer 43

but i can't show that lim ax^n=ac^n yet so i just leave it at that?

Answer 44

yea, you can... hmm. give me a min, read this for me http://tutorial.math.lamar.edu/Classes/CalcI/LimitsProperties.aspx

Answer 45

our goal will be to ONLY use those properties in order to manipulate the polynomial into something we can do

Answer 46

And we will be able to assume lim x=c and lim k=k

Answer 47

but that is it

Answer 48

i see ty for the help

Answer 49

np, if you want to try writing that up, I can check it, but this is something where you may not have enough info in class to answer

Answer 50

It all depends on those properties I linked to