# Very simple. What is wrong with this chain of limit equalities?

I know the first line is true and the last line is false. I don't know why the reasoning in between is wrong. $\lim_{x \rightarrow \infty} x^2 = \lim_{x \rightarrow \infty} x$ (True)

$((\lim_{x \rightarrow \infty} x^2)/(\lim_{x \rightarrow \infty} x)) = 1$

$\lim_{x \rightarrow \infty} (x^2/x) = 1$

$\lim_{x \rightarrow \infty} x = 1$ (false)

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Please, state things clearly. Is it $\displaystyle \lim_{x\rightarrow \infty}=c$, where $c$ is a real number, or is it $\displaystyle \lim_{x\rightarrow c}=\infty$?, where $c \in [-\infty, + \infty]$? –  Aloizio Macedo Feb 24 '13 at 22:19

To get to the 2nd line, you "divide both sides by infinity". That's a no-no.

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Yes, thank you. I certainly recognize dividing by infinity is not valid. I had to see that the limit operation is performed (giving infinity) before the division operation is performed. Thanks! –  David Feb 24 '13 at 22:26
Look at $\lim_{x \rightarrow \infty}(x^2/x)$. Cannot this be reduced? To $\lim_{x \rightarrow \infty}(x/1) = \lim_{x \rightarrow \infty}(x)$. Then its obvious that it does not tend to $1$ for the same reason the last one doesn't.