Unit 1: Limits & Continuity AP Calculus AB
Study Guide: Limits and Continuity
AP Calculus AB – Unit 1
1 Existence of Limits
For a limit to exist as x approaches c, the left-hand limit must equal the right-hand limit.
Definition of a Limit Existence
lim f (x) = L ⇐⇒ lim f (x) = lim f (x) = L
x→c x→c− x→c+
• If the left and right limits are different numbers, the limit Does Not Exist (DNE).
• If the function approaches ±∞, the limit technically does not exist, but we describe the
behavior as ∞ or −∞.
2 Calculating Limits Algebraically
2.1 Direct Substitution
Always try this first! If f (x) is continuous at c, then:
lim f (x) = f (c)
x→c
2.2 Indeterminate Forms ( 00 )
If substitution yields 00 , there is a ”hole” (removable discontinuity). You must manipulate the
function to find the limit.
1. Factoring: Factor numerator and denominator, cancel common terms, then substitute.
x2 − 9 (x − 3)(x + 3)
lim = lim = lim (x + 3) = 6
x→3 x − 3 x→3 x−3 x→3
2. Conjugates: Used for square roots. Multiply numerator and denominator by the conjugate.
√ √
x+4−2 x+4+2
lim ·√
x→0 x x+4+2
3. Trig Identities: Simplify using identities (e.g., sin2 x + cos2 x = 1).
2.3 Special Trigonometric Limits
Memorize these two key limits (as x → 0):
sin x 1 − cos x
lim =1 lim =0
x→0 x x→0 x
1
Study Guide: Limits and Continuity
AP Calculus AB – Unit 1
1 Existence of Limits
For a limit to exist as x approaches c, the left-hand limit must equal the right-hand limit.
Definition of a Limit Existence
lim f (x) = L ⇐⇒ lim f (x) = lim f (x) = L
x→c x→c− x→c+
• If the left and right limits are different numbers, the limit Does Not Exist (DNE).
• If the function approaches ±∞, the limit technically does not exist, but we describe the
behavior as ∞ or −∞.
2 Calculating Limits Algebraically
2.1 Direct Substitution
Always try this first! If f (x) is continuous at c, then:
lim f (x) = f (c)
x→c
2.2 Indeterminate Forms ( 00 )
If substitution yields 00 , there is a ”hole” (removable discontinuity). You must manipulate the
function to find the limit.
1. Factoring: Factor numerator and denominator, cancel common terms, then substitute.
x2 − 9 (x − 3)(x + 3)
lim = lim = lim (x + 3) = 6
x→3 x − 3 x→3 x−3 x→3
2. Conjugates: Used for square roots. Multiply numerator and denominator by the conjugate.
√ √
x+4−2 x+4+2
lim ·√
x→0 x x+4+2
3. Trig Identities: Simplify using identities (e.g., sin2 x + cos2 x = 1).
2.3 Special Trigonometric Limits
Memorize these two key limits (as x → 0):
sin x 1 − cos x
lim =1 lim =0
x→0 x x→0 x
1
