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Math Help - Find a and limit?

  1. #1
    Super Member fardeen_gen's Avatar
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    Find a and limit?

    If \lim_{\alpha\rightarrow \infty} \frac{\alpha^2 + 1}{\alpha + 1} - a\alpha exists, find a\in \mathbb{R} and the limit.

    EDIT: It has to be \alpha\rightarrow \infty. Nice book I have *Dripping in sarcasm* And I agree with the finite part of course.
    Last edited by fardeen_gen; May 11th 2009 at 10:42 PM. Reason: Fixed typo
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  2. #2
    MHF Contributor

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    Quote Originally Posted by fardeen_gen View Post
    If \lim_{n\rightarrow \infty} \frac{\alpha^2 + 1}{\alpha + 1} - a\alpha exists, find a\in \mathbb{R} and the limit.
    i think you meant \alpha \to \infty and not n \to \infty. also by "exists" you mean "being a finite number". to me if the limit is infinity, then it still exists. if the one sided limits are not equal,

    then the limit doesn't exist. other cases that i'd say the limit doesn't exist, are limits like: \lim_{x \to\infty} \sin x or \lim_{n\to\infty} (-1)^n, etc. enough about lecturing you! haha

    to answer your question, we have \frac{\alpha^2 +1}{\alpha + 1} - a \alpha = \frac{(1-a)\alpha^2 + 1 - a \alpha}{\alpha + 1}. it's clear now that the limit would be infinity unless a=1, and for a=1 the limit is -1.
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