limit evaluation

- anonymous

limit evaluation

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- anonymous

\[\lim_{x \rightarrow -4}\frac{ \sqrt{x^9+9}-5 }{ x+4 }\]

- myininaya

Try rationalizing the top by multiply the numerator and denominator by the top's conjugate

- anonymous

I worked it out to \[\frac{ x^9-16 }{ x+4(\sqrt{x^9+9}+5) }\]

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## More answers

- anonymous

I dont know what I would expand the top to

- anonymous

@myininaya

- myininaya

and i think you mean to have parenthesis around the (x+4) too on bottom

- anonymous

yes

- myininaya

try factoring the top to see if there is a factor of x+4 in it so you can get rid of what would make the bottom 0

- anonymous

I dont know how I would factor the top though and also with it being ^9 it's still a negative square root

- myininaya

do you know how do synthetic division or long division?

- anonymous

I should but I dont remember

- myininaya

ok well if you did know how and you determine there was no remainder then the limit does exist
but if there is a remainder then the limit will not exist
but i guess you can also determine this from just pluggin in -4 on both top and bottom if you have 0/0 the limit will exist
if you don't then it won't

- myininaya

if the limit does exist then you of course must find it
but if the limit doesn't exist, then you are done and dne is your answer

- anonymous

the limit does exist I know that, and I went back to my algebra 2 notes and doing synthetic division (x+4) is not a factor

- myininaya

how do you get the limit exists?

- anonymous

This is part of a review and when I put DNE, I was told I was incorrect

- myininaya

This question doesn't have a real limit had the function only exist for values where x^9+9 is greater than or equal to 0 anyhow
x^9 has to be greater to 0
which means x cannot be negative because if it makes since to say
if x is negative then x^9 is definitely more and most negative

- myininaya

sense not since*

- myininaya

you can even see a numerical approach will also show nothing exists to the left and right of -4 or at -4 (even though we aren't really concerned what happens at -4)

- myininaya

http://www.wolframalpha.com/input/?i=y%3D%28sqrt%28x%5E9%2B9%29-5%29%29%2F%28x%2B4%29+real
of course there are some negative values the function does exist for after all x^9+9 has to be greater than or equal to 0

- myininaya

i misphrased it a little above when i just said x^9 has to be greater than or equal to 0
only small negative values will work anyhow is what i mean

- myininaya

as you can also tell from the graph nothing exists around x=-4

- myininaya

the limit does not exist
we have determine this algebraically, graphically, and numerically

- myininaya

\[\lim_{x \rightarrow -4}\frac{\sqrt{x^9+9}-5}{x+4}\]
just to be certain this was the problem right?

- anonymous

yes

- myininaya

so what is needed to convince you this limit does not exist?
like which explanation does not make sense?

- anonymous

It's not that none of the explanations don't make sense. I came to the same conclusion working it myself. It's just that when I plugged in my answer what the test told me was that, that answer is incorrect

- myininaya

we can try to determine where the written went wrong in writing the test

- myininaya

give me a second and let me see if i can find out where they went wrong in writing the problem
because i bet you that is what happened

- myininaya

writer* not written

- anonymous

could I ask you one other problem real quick first

- myininaya

so if they are saying the limit exist then that means they want some (x+4)'s to cancel...
so one second

- myininaya

ok ask

- anonymous

http://www.wolframalpha.com/input/?i=%28x%2B1%29%2F%281%2B%281%2F%28x%2B1%29%29%29
For the domain why is it x=/=2 and x=/=-1

- anonymous

-2*

- anonymous

I got the -2 but not why -1 also

- myininaya

do me a favor while i look at that
try to answer that same question as if you were answering
\[\lim_{x \rightarrow -4}\frac{\sqrt{x^2+9}-5}{x+4}\]

- myininaya

i think you might get it right if you pretend that is the question

- anonymous

alright

- myininaya

let me know
and if you get it wrong i'm out of guesses of what they meant

- myininaya

ok you have a fraction inside that fraction

- myininaya

that one fraction does not exist at x=-1

- myininaya

and then after that of course you already knew to find when 1+1/(x+1) is not 0
which is when x=-2

- myininaya

for example when pluggin in -1 we get
\[\frac{1}{1+\frac{1}{-1+1}}=\frac{1}{1+\frac{1}{0}} \]
but 1/0 is not a number

- anonymous

alright so for \[\frac{ \sqrt{x^2+9}-5 }{ x+4 }\] I get\[\frac{ (x-4) }{ \sqrt{x^2+9}+5 }\] which still gives a negative root when I plug in -4

- myininaya

so 1/(1+1/0)) is still not a number

- myininaya

no it doesn't

- anonymous

sorry forgot ^2

- anonymous

so yeah never mind it gives 0

- myininaya

:(

- myininaya

ok you plugged in -4 right?

- myininaya

\[\frac{-4-4}{\sqrt{(-4)^2+9}+5}\]
I think you are getting tired of this problem and you aren't going through the arithmetic slowly enough

- myininaya

-4-4 is -8

- myininaya

and can you do the bottom?

- anonymous

\[\sqrt{16+9}=\sqrt{25}....... 5+5=10\] so 8/10 or 4/5

- myininaya

don't forget the negative sign from the top

- myininaya

so -4/5
tell me if that is what they were looking for or not

- anonymous

I don't know because it doesn't show me the correct answer, but it is something

- myininaya

you should let your teacher know of the mistake though

- myininaya

most teachers (or some teachers) are amazed by students who can find there errors

- myininaya

their*

- anonymous

I haven't met them yet, and I don't head to school until another week. What would be the most respectable non-annoying way to present it?

- myininaya

Definitely don't rub in there face. I would just bring it up once if and when they respond.
I think an email would be just fine.
You can be like:
Hey professor,
As I was taking the exam, I came across the problem lim (x->-4) (sqrt(x^9+9)-5)/(x+4) and I said the limit does not exist but it was returned incorrect by the exam.
I was thinking it was meant to be lim (x->-4) (sqrt(x^2+9)-5)/(x+4) where the answer would be -4/5.
Please let me know if I'm correct or not.
Sincerely,
(you)
But don't say she made a mistake.
Just say the exam or whatever.
What I'm saying is don't say YOU in the email. Try not to imply it is the teacher's fault even though it is. I think that is the best way.

- myininaya

Or dear professor if you are feeling more formal.

- anonymous

alright should I provide an explanation of my work about how I could not find the limit algebraically, graphically, or numerically for the original problem

- myininaya

I think you could say just nothing exists around -4 since x^9+9 has to be greater than or equal to 0.
You know which means x^9>=-9
But for x values less than -4 certainly do not satisfy the inequality x^9>=-9.
You could actually solve that inequality if you wanted to.
x>=(-9)^(1/9)
but values really close to -4 are definitely not in the set [(-9)^(1/9),inf)

- anonymous

alright thank you for all you help

- myininaya

Np. I don't have problem helping you with this one on your test mainly because it was written wrong. But I hope you intend to give the rest of the test your own knowledge.

- myininaya

Good luck @recon14193

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