Question

So I have this ques in limits, for example in (n^2+n)/ n^2 , and n approaches infinity. How is the answer : 1. Please need help.

Answer 1

when you want to find the limit of a polynomial as n approaches infinity you consider only the term of highest degree of the poly. and find its limit
so here you choose n2 in the numerator and n2 in the denominator
so you get n2 / n2 which is equal to 1

Answer 2

Another way to think of this kind of problem is: "What is the ending behavior" . As hoblos noted, this rational function has a horizontal asymptote of y = 1/

Answer 3

\( \lim \limits_{n \to \infty } \frac{n^2+n}{n^2} = \lim \limits_{n \to \infty } \frac{n^2}{n^2}+\frac{n}{n^2} = \lim \limits_{n \to \infty } (1+\frac 1 n) = 1\)

Answer 4

This is due to the fact that \( \lim \limits_{n \to \infty } \frac 1 n = 0\)

Answer 5

you are right FFM but sometimes you may need simple ways to solve an exercise
so if the function has more than two terms in the denominator you may face some difficulties and it will take more time solving it than just considering the highest degree term

Answer 6

I agree, the highest term trick works fast in complicated cases.

Answer 7

FFM is right of course (as always!), and so is hoblos; when taking the limit as the variable tends to infinity, only the highest powers need be considered. For example:
\[\Large \mathop {\lim }\limits_{x \to \infty } \frac{{{x^4} + {x^2} - 6823177x}}{{{x^3} - {x^4}}} = \mathop {\lim }\limits_{x \to \infty } \frac{{{x^4}}}{{ - {x^4}}} = \frac{1}{{ - 1}} = - 1\]This is because as x approaches infinity - i.e. the numbers are ridiculously big - any other number by comparison is going to become insignificantly small. 1000000000^4 = 10^36, 1000000000^2 = 10^18; you can see the difference is substantial already and that's hardly at the biggest numbers that will be reached.
In the last calculus unit I did, some time ago, we were told that it is fine to disregard everything but the highest powers, AS LONG AS we made a note of this - i.e. explicitly stated "Ignoring all but highest powers" - otherwise the equations become technically incorrect, since:\[\Large \frac{{{x^4} + {x^2} - 6823177x}}{{{x^3} - {x^4}}} \ne \frac{{{x^4}}}{{ - {x^4}}}\]

Answer 8

Please note this method only works while f(x) is a polynomial or rational polynomial, else it's won't work.

Answer 9

There are actually nice things you can do with limit tends to infinity, say compute this limit \[ \large\lim \limits_{n \to \infty} \frac{2^{b/n}+2^{2b/n}+2^{3b/n}+\cdots+ 2^{b}}{n} \]

Answer 10

As \[\lim1/n_{n \rightarrow \infty}=0\] the answer is 1 for the given question.
Taking n^2 common in the numerator we will end up with n^2(1+1/n) and then cancelling n^2 in numerator and denominator(Keep in mind this is true only if \[n \neq0\]) we get the said answer ie,1

Answer 11

Thank you so much guys! This will definitely help me in my finals.
BRAVO! :D