Question

lim [1/(x+3)4]=infinity. Prove
x→−3

Answer 1

I don't understand limits

Answer 2

CORRECTION. It should be lim as x approaches -3 [1/ (x+3)^4) = infinity

Answer 3

is this
\[\lim_{x\rightarrow -3}\frac{1}{(x+3)^4}\]?

Answer 4

YES and then it equals infinity

Answer 5

do you want to understand why it is true?

Answer 6

yes. like my teacher writes some statement using epsilon and delta.

Answer 7

then proves it.

Answer 8

cause 1/0 is infinity

Answer 9

ok before we worry about that, we know that x cannot be equal to -3 because you cannot divide by 0. so suppose x is say -2.9
then
\[-2.9+3=.1\] and
\[.1^4=.0001\] and finally
\[\frac{1}{.0001}=10000\]

Answer 10

just for the record
\[\frac{1}{0}\] is not a number and it is not infinity either

Answer 11

he doesnt jsut say that 1/0 is infinity. He will write liek for any M is greater than 0, we want delta greater tahn 0 such that then writes some statements with absolute values. At teh end, teh satement shud be like choose delta = ??

Answer 12

in any case what we see is that as x gets close to -3,

\[\frac{1}{(x+3)^4}\] will grow bigger and bigger

Answer 13

I'm confused how to do teh same thing with this problem. I'm not sure if anyone else learned it this way.

Answer 14

yes i understand what you want, i just wanted to give an example. in my example N was 10000 and delta was 0.1

Answer 15

oh ok.

Answer 16

suppose you want it to be bigger than say 1,000,000,000
what delta would you choose?

Answer 17

delta = 10000

Answer 18

hmmm i see we need some work here. delta is supposed to be the distance between x and -3
i certainly am not going to pick a very large number. it will be a very small number. like in my example i had N at 10000 and delta turned out to be only 0.1
so we expect a nice small delta. the bigger N, the smaller delta will be

Answer 19

that is the larger the output, the closer x will have to be to -3

Answer 20

lets solve it for N =1,000,000,000

Answer 21

yeah i mean when i insert the -3 into the equation i find 1 div 0 that cant solve so we have to insert value approx -3 in order to get finit

Answer 22

i want
\[\frac{1}{(x+3)^4}>1,000,000,000\]

Answer 23

to solve this inequality for x, i would take the reciprocal of both sides and write
\[(x+3)^4<0.000 000001\]

Answer 24

then take the 4th root of both sides to get
\[-.01<x+3<.01\]

Answer 25

in absolute value notation this says
\[|x+3|<0.01\]

Answer 26

good explanation satellite

Answer 27

thx

Answer 28

so to get what you need, replace 1000000000 by N and the find the appropriate delta
the steps are identical except with N instead of the number

Answer 29

start with
\[\frac{1}{(x+3)^4}>N\]
then
\[(x+3)^4<\frac{1}{N}\] then

\[-\frac{1}{\sqrt[4]{N}}<x+3<\frac{1}{\sqrt[4]{N}}\]
finally
\[|x+3|<\frac{1}{\sqrt[4]{N}}\]and you are done

Answer 30

So how do i write that for any N greater than 0, we want delta greater tahn 0 such taht ...then use absolute value statements?

Answer 31

For ex. a prob we did was lim as x approaches 0, 1/x^2 = infinity. So the statement was for any M greater than 0, we want delta greater tahn 0 so taht absolute value of x is less than delta implies that 1/x^2 is greater tahn M.

Answer 32

Do u understand what i mean??This is what i wrote but as an eqation

lim 1/x^2 = infinity.
x app. 0

For any \[M > \delta\] we want \[\delta > 0\] so that \[\left| x \right| < \delta \implies that 1/x ^{2} >M\]
\[1/X ^{2} > M \implies x ^{2} < 1/M \implies \sqrt{x ^{2}} < \sqrt{1/M} \]
\[\left| X \right| < 1/\sqrt{m} \]
Choose \[\delta = 1/\sqrt{M}\]

Answer 33

yes it is what i wrote above exactly, but i used N instead of M and found
\[\delta =\frac{1}{\sqrt[4]{N}}\] would do

Answer 34

btw for when you say the limit it is infinity you do not usually say "for N > 0" because you are concerned with large N. you usually say "for any N no matter how large" or something like that

Answer 35

oh okay. Is this correct??
For any for any N no matter how large, we want delta greater tahn 0, so taht
\[\left| x+3 \right| <\delta\] .....etc (what u wrote)