In finding the values of limits, some times we obtain the following forms , 0 × ∞, ∞ – ∞, 0⁰, 1^∞, ∞⁰ which are not defined.

Some standard results on Indeterminate forms

i) If n is a rational number, then = n.a^{n – 1}

ii) If m, n are two real numbers, then a^{m – n}

2. Trigonometric limits (where x is measured in radians)

i) sin x = 0	ii) sin x = sin a, ∀ a ∈ R
iii) cos x = 1	iv) cos x = cos a, ∀ a ∈ R
v) tan x = 0	vi) tan x = tan a, ∀ a ≠ (2n + 1); n ∈ Z
vii) = 1 =	viii) = k, ∀ k ∈ R
ix) = 1 =	x) = k, ∀ k ∈ R
xi) = 1 =
xii) = 1 =
xiii) = = 0
xiv) = 1	xv) = 0
xvi) and both does not exist
xvii) sin–1x = sin–1a; | a | ≤ 1
xviii) cos–1x = cos–1a; | a | ≤ 1
xix) tan–1x = tan–1a; – ∞ < a < ∞
xx)

3. Exponential limits

i) = 1	ii) = k; k ≠ 0
iii)	iv) ; k ≠ 0
v) ; a, b > 0

4. Miscellaneous

i) If f(x) = g(x) = 0, then	=
ii) If f(x)	=	1 and g(x) = ∞, then
	=	[1 + f(x) – 1]g(x)
	=	{(1 + f(x)) – 1}g(x)
	=

iii) = e	iv) = eλ
v) = e	vi) = eλ
vii) = e	viii) = eλ
ix) = epq
x) = epq
xi) ax =

5. Logarithmic limits

i. = 1	ii. = p
iii. = – 1	iv. = – p
v. , a > 0, ≠ 1