#Infinite limits

167 messages · Page 1 of 1 (latest)

gritty chasm
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Can someone explain limits at infinite?

spring leafBOT
lime sorrel
gritty chasm
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I know that higher exponent on top (than the bottom) means it’s +/- inf

lime sorrel
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ohh

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ok

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Basically you can consider three cases

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The first is what you mentioned

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Basically when the function in the numerator grows faster than in the denominator then as x approaches to infinity so does the limit

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Basically in polynomials when the degree in the numerator is higher than in the denominator then that is the case

gritty chasm
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okay but what if it’s something like 6x^3/3x^2?

lime sorrel
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what about it

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what do you think happens

gritty chasm
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it approaches positive infinity

lime sorrel
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yes

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also

gritty chasm
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do you not divide 6/3?

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in what cases do you divide the coefficients

lime sorrel
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6x^3/3x^2 = 2x

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and 2x obv goes to inf

gritty chasm
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ok

lime sorrel
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only then

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the quotient of coeffiecients is the limit

gritty chasm
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oh

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what if the numerator exponent is lesser than the denominator

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is it 0

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or something

lime sorrel
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,,\lim_{x \to \infty} \frac{a_0x^n + a_1x^{n-1} +... + a_n}{b_0x^n + b_1x^{n-1} + ... + b_n} = \frac{a_0}{b_0}

terse bobcatBOT
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𝔸dωn𝓲²s

587715738484211754.png
lime sorrel
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The reason why this works is

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if you factorize the dominant term x^n

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everything with lower degree will go to 0

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and what remains is a_0/b_0

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does that make sense?

lime sorrel
gritty chasm
lime sorrel
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basically there is a theorem for this

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but to prove it to yourself

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consider factorzing always the dominant term

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and see what happens

gritty chasm
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but wait do the signs of the coefficient determine if it’s positive or negative infinite right?

lime sorrel
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yes

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the sign of the dominant term

gritty chasm
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ok

lime sorrel
gritty chasm
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I think those are the basics but how does limits at infinite work on stuff like 1/x or e^x, or e^-x?

lime sorrel
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basically you ask yourself what happens with the function as x grows bigger

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and when you have something like this

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,,\lim_{x \to \infty} \frac{x^{100}}{e^x}

terse bobcatBOT
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𝔸dωn𝓲²s

587715738484211754.png
lime sorrel
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Then you are thinking in terms of which infinity is grows bigger and faster

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and you can keep in mind any exponential function will always dominate any polynomial

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if x goes to infinity

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this means

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e^x at some point becomes much bigger than x^100

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so it eventually resemble something like ~ 1/x which means it goes to 0 here

gritty chasm
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sorry but how does that work on like 1/x? if x at some point becomes larger than 1

lime sorrel
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i meant

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the limit behaves like 1/x

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which means

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x^100 is so small in comparison to e^x

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like 1 is very small in comparison for x

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for very big x

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i could also take

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100000000000/x

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still that would go to 0

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eventually

gritty chasm
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is this kind of like the other rule where if the numerator is small than the denominator it equals 0?

lime sorrel
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yes

gritty chasm
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oh that makes sense

lime sorrel
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,w plot x^(100)/e^x between 0 and 700

terse bobcatBOT
lime sorrel
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ok the y-axis is messed

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but basically

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at some big x

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like here 200 or 600

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x^100 become relatively small to e^x

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like 1 would to x

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ratio wise

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,w plot y = 1/x between 0 to 700

terse bobcatBOT
lime sorrel
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this is what i meant by behave the same

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that they go both to 0

gritty chasm
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okay and for (x^2 + 3x)/(2x - 5), how would you solve this if x -> inf

would it be positive infinite since ^2 > ^1? and since there’s no negative symbol, I’m just making sure

lime sorrel
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yes

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that's it

gritty chasm
lime sorrel
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if you have to show it

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no

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thats tricky but

gritty chasm
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oh

lime sorrel
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the numerator would go infinity and dominate the denominator again yes

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how ever the denominator becomes negative

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so it would be -infinity

gritty chasm
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do you ever need to integrate or find the derivative when a limit of x approaches +/- infinity?

lime sorrel
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idk about integration but for limits there exists L'Hopital where you take the derivative separetly from numerator and denominator when you have cases like 0/0 or inf/inf (notice -inf/inf or inf/-inf is also allowed)

gritty chasm
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ok and last thing when the powers are equal you subtract their coefficients right

lime sorrel
gritty chasm
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oops I meant divide

gritty chasm
lime sorrel
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if you deal with polynomials you can do that

gritty chasm
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ok

lime sorrel
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that technically would work in cases like this too

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,,\lim_{x \to \infty} \frac{4e^{2x}+x^2}{3e^{2x}+e^x+\ln(x)}

terse bobcatBOT
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𝔸dωn𝓲²s

587715738484211754.png
lime sorrel
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it's be 4/3

gritty chasm
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so I’m assuming the limit 4/3

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yea

lime sorrel
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yes

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but still be careful

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you can use this more of like what you expect the outcome to be

gritty chasm
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ok when the numerator's power is lesser than the denominator, the limit always equals 0 right?

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idk if it’s alwayssss but most of the time?

lime sorrel
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here you would factorize e^(2x) and then you can argue that an exponential dominates a polynomial and more any logarithm

lime sorrel
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more of

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not lesser

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you mean

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if it's slower growing

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basically if the numerator just cant keep up with the speed

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of the denominator

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if that makes sense

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imagine it like a race of functions

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the faster growing dominates at infinity

gritty chasm
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what if you have x -> inf for something like 4x^2/2? would it just be infinite

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positive

lime sorrel
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well yes, why are you doubting this simple example?

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also you can try to simplify before considering the limit

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to make it more obvious

gritty chasm
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I just don’t wanna get easy and simple ones like those wrong

lime sorrel
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well it's a parabola

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clearly any parabola goes to infinity

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depending on the sign of course

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just ask yourself

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what happends with your function as x goes bigger

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and then draw a conclusion

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and most functions we know how they behave

gritty chasm
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in like a basic polynomial expression like 5x^6 - 2x^3 - 3, would you primarily focus on the 5x^6, and would you focus on the operation of 5, or operation of 6?

lime sorrel
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yes

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x^6 is the dominant term

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and the coeffiecent 5 is positive

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hence +inf

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done

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the rest can go to hell haha

gritty chasm
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what if it was -5x^6?

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😭

lime sorrel
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then you basically reflected the function

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logically

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at the x-xis

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so now it goes to -inf

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x^6 is the dominant again

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coefficient is negative done

gritty chasm
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yea and I’m assuming if it was like 5x^-6, you would focus on the 2x^3?

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it’s - 2x^3 so -inf?

lime sorrel
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if you had 5x^(-6) - 2x^3 - 3

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yes

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dominant term

gritty chasm
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ok I think I understand everything

lime sorrel
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x^(-6) goes to 0

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so it's even less impacting

gritty chasm
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okay thank you for your time and help I think I’m done now

lime sorrel
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very cool

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.solved