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I am trying to apply a limit law in which the limit is distributed to both the numerator and the denominator. However, the \lim_{x\to \infty} does not render properly when I place it in a fraction (\frac{\lim_{x\to \infty}}{\lim_{x\to \infty}}). The _{x\to \infty} part shows up next to the lim.

Any help would be greatly appreciated.

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Either

$\frac{\displaystyle\lim_{x\to \infty}}{\displaystyle\lim_{x\to \infty}}$

or

$\frac{\lim\limits_{x\to \infty}}{\lim\limits_{x\to \infty}}$.
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    I'd prefer the second form, but such a big fraction should go in a display: if placed inline it will completely spoil the page. – egreg Apr 14 '13 at 21:52
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    @PrzemysławScherwentke: Note that using such a definition, all subsequent math content will be set using \displaystyle (not just the \dfrac. To get around this, you need to grab the arguments and contain the \displaystyle inside a group. – Werner Apr 14 '13 at 22:31

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