In finding the values of limits, some times we obtain the following forms
, 0 × ∞, ∞ – ∞,
0^{0}, 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.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^{–1}x = sin^{–1}a; | a | ≤ 1 | ||
xviii) cos^{–1}x = cos^{–1}a; | a | ≤ 1 | ||
xix) tan^{–1}x = tan^{–1}a; – ∞ < 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) = e^{λ} |
v) = e | vi) = e^{λ} |
vii) = e | viii) = e^{λ} |
ix) = e^{pq} | |
x) = e^{pq} | |
xi) a^{x} = |
5. Logarithmic limits
i. = 1 | ii. = p |
iii. = – 1 | iv. = – p |
v. , a > 0, ≠ 1 |