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In my coursebook I am practising questions which ask what the value is when the variable, n tends to infinity. Unfortunately there are no answers. Although I have solved I think correctly.

My question in general is are my questions correct and Iam I using the right method? By subbing the values into my graphical calculator I appear to have the correct solution I feel.

My method is largly dividing by the highest order variable in the expresssion.

  1. (n+1)/(1+n^2)

Divide by largest power n^2:

This is: (1/n + 1/n^2) / (1 + (1/n^2) = 0/1 = 0

  1. (1+n^2) / (1+n)

Divide by largest power n^2:

That is: (1/n^2 + 1) / (1/n^2 + 1/n) = (0+1)/(0+0) = 1/0 = infinity

  1. (2n+1)/(1+n^3)

Divide by largest power n^3:

That is: (2/n^2 + 1/n^3) / (1 + 1/n^3) = (0/1) = 0

  1. (-1 + 3n^2) / (1+n^2)

Divide by largest power n^2:

That is: (-1 / n^2 + 3n^2 / n^2) / (1/n^2 + n^2 / n^2)

3/1 = 3

  1. (n+1)/(1+2n)

Divide by largest power n:

(n/n + 1/n) / (1/n + 2n/n) = (1+0) / (0+2) = 1/2

  1. (4+n^2) / (-2 + 3n)

Divide by largest power n^2:

(4/n^2 + n^2/n^2 ) / (-2/n^2 + 3n/n^2) = (0+1)/(0+0) = 1/0 = infinity

Thank you!

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    If anyone could tell me how to change to latex it would be great thanks – 2017-02-01
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    It's a better notation to use $\to$ instead of $=$ when passing to the limit, by the way. – 2017-02-03
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    Looks pretty good. This is the method. That I would use, is it possible to latex or make it easier to read please :) – 2017-02-03

1 Answers 1

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Yep, when the variable tends to infinity, any polynomial behaves like its term of the highest degree, as it grows faster than all others.

For a rational fraction, the behavior is the same as that of the ratio of the terms of the highest degree, hence like $\frac{a_n}{b_d}x^{n-d}$.