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list of common limits∗
Wkbj79†
2013-03-22 0:05:31
Following is a list of common limits used in elementary calculus:
• For any real numbers a and c, lim c = c.
x→a
• For any real numbers a and n, lim xn = an (proven here for n a positive
x→a
integer)
sin x
= 1 (proven here)
x→0 x
• lim
• lim
1 − cos x
= 0 (proven here)
x
• lim
arcsin x
= 1 (proven here)
x
x→0
x→0
ex − 1
= 1 (proven here)
x→0
x
• lim
ax − 1
= ln a (proven here).
x→0
x
• For a > 0, lim
xa
= 0 (proven here).
x→∞ bx
• For b > 1 and a any real number, lim
• lim xx = 1 (proven here)
x→0+
• lim+ x ln x = 0 (proven here)
x→0
• lim
x→∞
ln x
= 0 (proven here)
x
1
• lim x x = 1 (proven here)
x→∞
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1
•
x
1
1+
=e
x→±∞
x
lim
1
• lim (1 + x) x = e
x→0
1
• lim (1 + sin x) x = e (power of e, l’Hôpital’s rule)
x→0
• lim (x −
p
x→∞
x2 − a2 ) = 0 (proven here)
• For a > 0 and n a positive integer, lim
x→a
• lim
x→0
1
x−a
=
.
xn − an
nan−1
tan x − sin x
1
=
(by l’Hôpital’s rule)
x3
2
(log x)p
=0
x→∞
xq
π
tan x + tan ξ
sec2 ξ
• tan x +
= limπ
= − cot x (by
= limπ
ξ→ 2 − tan x sec2 ξ
ξ→ 2 1 − tan x tan ξ
2
l’Hôpital’s rule)
That is, tan x tan(x+ π2 ) = −1, which indicates orthogonality of the slopes
represented by those functions.
• For q > 0, lim
• For a real or complex constant c and a variable z,
nn+1 n −(n+1)
lim n+1 c +
= e−cz .
n→∞ z
z
√
• For x real (or complex), limn→∞ n( n x − 1) = log x (proven here for real
x).
Feel free to add! Also, if the limit you decide to add is proven somewhere
on PlanetMath, please provide a link. Thanks.
References
[1] Catherine Roberts & Ray McLenaghan, “Continuous Mathematics” in Standard Mathematical Tables and Formulae ed. Daniel Zwillinger. Boca Raton:
CRC Press (1996): 333, 5.1 Differential Calculus
2