how to evaluate this limit

fran1942

Hello, I am just starting to learn limit evaluation techniques. I am unsure of the method used in this case.

(x^3 + x + 2) / (x^4 -x +1)
limit tending towards infinity.

I know the first step is 'x^3 / x^4', then '1/x = 0'
But I dont understand how this came about.

Can someone please clarify these steps.
Thanks kindly if possible.

pwsnafu

You want a proof of the limit of 1/x is 0?
Do you know the definition of limit?

Another thing. "1/x = 0" is wrong its "1/x -> 0 as x -> infinity".

Mark44

Factor x⁴ out of every term in both the numerator and denominator, and then take the limit.

Zatara7

normally teachers don't really look for the worked out solution.. they just expect you to know if it is 0 or infinity or just the coefficients. and that all depends on the greatest power of the denominator and numerator

GDLY

Basically you can just factor everything by x^4 like Mark44 says but then you'd have to factor it out of the constants too and you'll end up with a jumbled mess, which if you then evaluate the limit for you can prove is 0.

It's important for you to be able to conceptualize this and understand why it approaches 0.

The degree of the exponent in the bottom is larger than any in the top, therefore, infinity climbs faster at the bottom than it does at the top and at infinity the rate is infinitely greater thus the numerator eventually becomes insignificant.

qazi75

Well come.Please see these steps carefully.
(lim x...>∞(x3+x+1/x4-x+1)
taking x3 and x4 common 4m numerator & denominator respectively
=lim x...>∞[x3(1+1/x2+1/x3)/x4(1-1/x3+1/x4)]
x3 and x4 will cancel each other which comes 1/x
=lim x...>∞[1(1+1/x2+1/x3)/x(1-1/x3+1/x4)]
as by applying limit,1/x2 comes 1/(∞)2=1/∞=0 and so on,so
=1(1+0+0)/∞(1-0+0)
=1/∞
=0.