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