Find limit at Infinity | sqrt(1+4x^6)/(2-x^3) x approaches infinity | general solution

To calculate the limit at infinity, we need to compare the highest power of denominator and numerator.
There are in general 3 types:
1. highest power (after simplify) of denominator greater than numerator, the limit often goes to 0.
2. highest power of dominator and numerator are the same, the limit is the highest term coefficient.
3. highest power of denominator smaller than numerator, the limit often goes to infinity (or negative infinity).

[@childrenofkoris]

you're a genius, i tested by substituting the value of x to 10,000 integer and its came close to value of -2 , so basically any limits related to infinity, with any variables will only involve the power degree of the D and N from the expression, and other constants can be omitted, pure genius :) i love it

[@simohayha6031]

Theory says that if lim x --> a f(x) equals b, then lim x --> a (f(x))^2 equals b^2. So we can square f to get a simplified function of the form p(x)/q(x) where p, q are polynomials. This should trigger you, as polynomials simplify to constants when taking the derivative. So you can just start applying L'hôpital's rule however many times you need and doing this will give 4. Since this 4 represents the b^2, our answer is sqrt(4) = +/- 2. Sign analysis shows that in f, if x goes to infintiy we get inf/(-inf), so the -2 is the correct answer.
Hence lim x --> infinity of f(x) = -2
Its an amazing way of getting rid of square roots.
*Edit why the fk is my text crossed out

[@Idkwhattonamess]

I was confused at first, but when you explained how to simplify with the 2x^2 + 5/3x^3 + 2x + 3 example, it made so much more sense

[@boguslawszostak1784]

After dividing the numerator and denominator by the highest power of x in the denominator:
If the highest power in the numerator is greater than 0, the limit as x→∞ approaches infinity (+∞ or −∞), depending on the signs of the term.
If the highest power in the numerator is equal to 0, the limit as x→∞ exists and is finite and easy to calculate.
If the highest power in the numerator is less than 0, the limit as x→∞ approaches zero.

[@Svabby]

Don't you want to divide everything by the highest exponent from the denominator? So x^3? For the first example

[@ZainAli-nk1rq]

Amazingly explained,thank you so much

[@shivalichainani5871]

You are a lifesaver

[@EricPham-gr8pg]

4
Because as we square up both parts
Then we get
(1+4x^6)/(4-4x3+ x^6)
As x get to near infinity we had ((4x^6)+1 )/(1x^6)
So 4 is nearly correct because 1/x^6 is too small

[@awusacollins]

Very helpful!!

[@panashepanaxe629]

Thank you mam Rebecca.well xplained

[@arslanmoghal5472]

Good explanation 👍

[@calcwithRebecca]

Thanks 🙂

[@somebody7270]

Thanks!

[@PatwariVivek.]

I like it so much ❤

[@Divine-Education-1980]

Which country

[@eoeoeoei8226]

❤❤thank youuu

[@bisiberh1451]

Thanks

[@paulwood3460]

👍👍👏👏

[@user-st3ej6bi9b]

-2

[@NurHadi-qf9kl]

1/2

[@PatwariVivek.]

❤😮😂😅hi

[@ghamoz]

Sono infiniti dello stesso ordine, immedito il limitr

[@chuckc3665]

you spent 8 minutes to do a problem that can be done in 8 seconds. you better not to upload videos any more

[@lopsyd2772]

tapohaaa

[@markowenrayos3658]

She didn't only solved the proble but also explained very well. So think before write your stupid comment.

[@chuckc3665]

she explained to the stupid@@markowenrayos3658

[@kornelviktor6985]

Dont be so toxic bro

[@abueghan1363]

Why man. She's not solving for you okay buh those who have challenges so you better think before write

[@user-st3ej6bi9b]

-2