- #1

late347

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## Homework Statement

f[/B]

f(x)= x^2 +4

find the limit as x approaches 1, there is something wrong with the latex code but I don't know what.

Limit $$\lim_{x\to 1} \frac{{f(x)}^4-{f(1)}^4}{x-1}$$

## Homework Equations

-methods for finding limits

-factorising polynomials

-possibly polynomial long division (I used it)

## The Attempt at a Solution

[/B]

Well, I was a little bit bored today and decided to try to do some "more difficult" problems in my textbook for limits. You are free to comment on my polynomial long division if you want to. I tried to use it for this problem. Because for rational function if you can cancel out the pesky denominator... then you can plug in the x= something and get the limit that way.The value of the rational clause comes out at 0/0 just something to note in the beginning. Existence of limit as x->1 is still possible. We cannot immediately plug in x=1 and get the value of the limit of the rational function.

But, we ought to try to cancel the clause of function f so we are able to plug in x=1

If the expression is not cancellable, then we are going to have to start tabulating values for plus-side limit and minus-side limit, and evaluate if the both-sided limit exists.

I did the problem the tedious and long way, but it seemed to bring the correct answer eventually.

I was wondering if there's any easier way (faster way, smarter way) to get the factorization correct. Well... you could put the expression into wolfram alpha and have the computer factor it.

Foiling out the expression with a little bit pen-and-paper calculation

##[(x^4+8x^2+16)^2 -625]/(x-1)##

=

foiling out more with pen-and-paper

##(x^8+16x^6+96x^4+256x^2-369)/(x-1)##

putting that into polynomial long division in the Anglo-American format of the long division.

After some time, the remainder comes out at zero (which is good) because it implies our original expression was indeed factorable.

pic of polynomial long division. I attempted to follow the instruction in the Finnish language wiki article as best as I could. Anglo_american long division is indeed taught as the only long division in most schools I reckon...

The quotient is ##(x^7+x^6+17x^5+17x^4+113x^3+113x^2+369x+369)##

and remainder =0

Hence we can continue and cancel the expression with (x-1) out of there##\lim_{x\to 1}\frac{[x-1](x^7+x^6+17x^5+17x^4+113x^3+113x^2+369x+369)}{x-1}##

##\lim_{x\to 1}{(x^7+x^6+17x^5+17x^4+113x^3+113x^2+369x+369)}##

plug in x=1 and we will have the limit = 1000, as x->1

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