Calculus: Indeterminate Forms & L’Hôpital’s Rule
1 Introduction
Indeterminate forms occur when direct substitution in a limit does not yield a clear answer
The 7 Indeterminate Forms
The standard indeterminate forms are:
0 ∞
, , 0 × ∞, ∞ − ∞, 1∞ , 00 , ∞0
0 ∞
2 L’Hôpital’s Rule
Condition: Use this rule specifically for the 00 or ∞ ∞ forms Theorem: If f (x) and g(x) are
differentiable functions and f (a) = g(a) = 0 (or both approach ±∞),
f (x) f ′ (x)
lim = lim ′
x→a g(x) x→a g (x)
3 Important Expansions (Taylor Series)
Using series expansions is often the fastest way to solve limits
(1-x)−1 = 1 + x + x2 + x3 + . . .
x2 x3
ex = 1 + x + 2! + 3! + ...
x3 x5
sin(x) = x − 3! + 5! − ...
x2 x4
cos(x) = 1 − 2! + 4! − ...
x3 2 5
tan(x) = x + 3 + 15 x + ...
x2 x3
log(1 + x) = x − 2 + 3 − ... (for |x| < 1)
Standard Limits
• limx→0 sin x
x =1
• limx→0 tan x
x =1
• limx→0 cos x = 1
1
1 Introduction
Indeterminate forms occur when direct substitution in a limit does not yield a clear answer
The 7 Indeterminate Forms
The standard indeterminate forms are:
0 ∞
, , 0 × ∞, ∞ − ∞, 1∞ , 00 , ∞0
0 ∞
2 L’Hôpital’s Rule
Condition: Use this rule specifically for the 00 or ∞ ∞ forms Theorem: If f (x) and g(x) are
differentiable functions and f (a) = g(a) = 0 (or both approach ±∞),
f (x) f ′ (x)
lim = lim ′
x→a g(x) x→a g (x)
3 Important Expansions (Taylor Series)
Using series expansions is often the fastest way to solve limits
(1-x)−1 = 1 + x + x2 + x3 + . . .
x2 x3
ex = 1 + x + 2! + 3! + ...
x3 x5
sin(x) = x − 3! + 5! − ...
x2 x4
cos(x) = 1 − 2! + 4! − ...
x3 2 5
tan(x) = x + 3 + 15 x + ...
x2 x3
log(1 + x) = x − 2 + 3 − ... (for |x| < 1)
Standard Limits
• limx→0 sin x
x =1
• limx→0 tan x
x =1
• limx→0 cos x = 1
1
