Indeterminate forms

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

Some standard results on Indeterminate forms

1. Algebraic limits

i) If n is a rational number, then = n.an – 1

ii) If m, n are two real numbers, then am – 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