Question

limit evaluation

Answer 1

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

Answer 2

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

Answer 3

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

Answer 4

I dont know what I would expand the top to

Answer 5

@myininaya

Answer 6

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

Answer 7

yes

Answer 8

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

Answer 9

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

Answer 10

do you know how do synthetic division or long division?

Answer 11

I should but I dont remember

Answer 12

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

Answer 13

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

Answer 14

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

Answer 15

how do you get the limit exists?

Answer 16

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

Answer 17

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

Answer 18

sense not since*

Answer 19

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)

Answer 20

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

Answer 21

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

Answer 22

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

Answer 23

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

Answer 24

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

Answer 25

yes

Answer 26

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

Answer 27

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

Answer 28

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

Answer 29

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

Answer 30

writer* not written

Answer 31

could I ask you one other problem real quick first

Answer 32

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

Answer 33

ok ask

Answer 34

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

Answer 35

-2*

Answer 36

I got the -2 but not why -1 also

Answer 37

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

Answer 38

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

Answer 39

alright

Answer 40

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

Answer 41

ok you have a fraction inside that fraction

Answer 42

that one fraction does not exist at x=-1

Answer 43

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

Answer 44

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

Answer 45

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

Answer 46

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

Answer 47

no it doesn't

Answer 48

sorry forgot ^2

Answer 49

so yeah never mind it gives 0

Answer 50

:(

Answer 51

ok you plugged in -4 right?

Answer 52

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

Answer 53

-4-4 is -8

Answer 54

and can you do the bottom?

Answer 55

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

Answer 56

don't forget the negative sign from the top

Answer 57

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

Answer 58

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

Answer 59

you should let your teacher know of the mistake though

Answer 60

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

Answer 61

their*

Answer 62

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?

Answer 63

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.

Answer 64

Or dear professor if you are feeling more formal.

Answer 65

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

Answer 66

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)

Answer 67

alright thank you for all you help

Answer 68

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.

Answer 69

Good luck @recon14193