infinity tricks

  table of limits

Infinity tricks are informal mathematical techniques with ∞ (infinity) to help evaluate limits and limits with integrals at or near infinity. Such "tricks" are often a first resort. When they do not avail, then l'Hpital's rule for the evaluation of indeterminate forms in many cases reduces limits to forms for which these "tricks" work. The symbol

\infty

does not represent a number; it represents a limit only. The expression N^∞ means

\lim_{x\to\infty}n^x,\,

and similarly in other cases. Some well-known limits>

b>Form	Condition	Answer	Notes
math>\lim_{x\to\infty}N/x	any real N	0
rowspan=3 \| $\lim_{x\to\infty}x/N$	any N > 0	$\infty$
i>N = 0	does not exist
ny N < 0	$-\infty$
rowspan=3 \| $\lim_{x\to\infty}x^N$	any N > 0	$\infty$
i>N = 0	1
ny N < 0	0
rowspan=3 \| $\lim_{x\to\infty}N^x$	any N > 1	$\infty$
i>N = 1	1
< N < 1	$0$
math>\lim_{x\to\infty}N^{-x}	any N > 1	0	$=1/N^\infty$
rowspan=3 \| $\lim_{x\to\infty}\sqrt x {N}$	N > 0	1
i>N = 0	0
ny N < 0	does not exist
rowspan=3 \| $\lim_{x\to\infty}\sqrt N {x}$	any N > 0	1
i>N = 0	(−1..1)
ny N < 0	−1
math>\lim_{x\to\infty}\log x	\| $\infty$ \|
math>\lim_{x\to0}\log x	\| $-\infty$ \|(provided 0 is approached from above)

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