Limits Cheat Sheet — All Calculus Limit Rules & Techniques A complete reference of calculus limit rules and techniques. Includes basic properties, special limits (trig, exponential), L'Hôpital's rule, indeterminate forms, infinity limits, and one-sided limits — everything you need for AP Calculus and Calculus I.
Basic Limit Properties Constant Limit
lim x → a c = c \lim_{x \to a} c = c x → a lim c = c Limit of x
lim x → a x = a \lim_{x \to a} x = a x → a lim x = a Sum Rule
lim x → a [ f ( x ) + g ( x ) ] = lim f ( x ) + lim g ( x ) \lim_{x \to a} [f(x) + g(x)] = \lim f(x) + \lim g(x) x → a lim [ f ( x ) + g ( x )] = lim f ( x ) + lim g ( x ) Difference Rule
lim x → a [ f ( x ) − g ( x ) ] = lim f ( x ) − lim g ( x ) \lim_{x \to a} [f(x) - g(x)] = \lim f(x) - \lim g(x) x → a lim [ f ( x ) − g ( x )] = lim f ( x ) − lim g ( x ) Constant Multiple
lim x → a c f ( x ) = c lim x → a f ( x ) \lim_{x \to a} c f(x) = c \lim_{x \to a} f(x) x → a lim c f ( x ) = c x → a lim f ( x ) Product Rule
lim x → a [ f ( x ) g ( x ) ] = lim f ( x ) ⋅ lim g ( x ) \lim_{x \to a} [f(x) g(x)] = \lim f(x) \cdot \lim g(x) x → a lim [ f ( x ) g ( x )] = lim f ( x ) ⋅ lim g ( x ) Quotient Rule
lim x → a f ( x ) g ( x ) = lim f ( x ) lim g ( x ) ( lim g ≠ 0 ) \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim f(x)}{\lim g(x)} \quad(\lim g \ne 0) x → a lim g ( x ) f ( x ) = lim g ( x ) lim f ( x ) ( lim g = 0 ) Power Rule
lim x → a [ f ( x ) ] n = [ lim x → a f ( x ) ] n \lim_{x \to a} [f(x)]^n = \left[\lim_{x \to a} f(x)\right]^n x → a lim [ f ( x ) ] n = [ x → a lim f ( x ) ] n Special Limits Sin x / x
lim x → 0 sin x x = 1 \lim_{x \to 0} \frac{\sin x}{x} = 1 x → 0 lim x sin x = 1 (1 − cos x) / x
lim x → 0 1 − cos x x = 0 \lim_{x \to 0} \frac{1 - \cos x}{x} = 0 x → 0 lim x 1 − cos x = 0 (1 − cos x) / x²
lim x → 0 1 − cos x x 2 = 1 2 \lim_{x \to 0} \frac{1 - \cos x}{x^2} = \frac{1}{2} x → 0 lim x 2 1 − cos x = 2 1 tan x / x
lim x → 0 tan x x = 1 \lim_{x \to 0} \frac{\tan x}{x} = 1 x → 0 lim x tan x = 1 Definition of e (Limit)
lim n → ∞ ( 1 + 1 n ) n = e \lim_{n \to \infty} \left(1 + \tfrac{1}{n}\right)^n = e n → ∞ lim ( 1 + n 1 ) n = e Exponential Limit
lim x → 0 e x − 1 x = 1 \lim_{x \to 0} \frac{e^x - 1}{x} = 1 x → 0 lim x e x − 1 = 1 Log Limit
lim x → 0 ln ( 1 + x ) x = 1 \lim_{x \to 0} \frac{\ln(1 + x)}{x} = 1 x → 0 lim x ln ( 1 + x ) = 1 Power Limit
lim x → 0 ( 1 + x ) n − 1 x = n \lim_{x \to 0} \frac{(1 + x)^n - 1}{x} = n x → 0 lim x ( 1 + x ) n − 1 = n L'Hôpital's Rule L'Hôpital (0/0 form)
lim x → a f ( x ) g ( x ) = lim x → a f ′ ( x ) g ′ ( x ) ( 0 0 ) \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} \quad\left(\tfrac{0}{0}\right) x → a lim g ( x ) f ( x ) = x → a lim g ′ ( x ) f ′ ( x ) ( 0 0 ) L'Hôpital (∞/∞ form)
lim x → a f ( x ) g ( x ) = lim x → a f ′ ( x ) g ′ ( x ) ( ∞ ∞ ) \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} \quad\left(\tfrac{\infty}{\infty}\right) x → a lim g ( x ) f ( x ) = x → a lim g ′ ( x ) f ′ ( x ) ( ∞ ∞ ) Convert 0 · ∞
f ⋅ g = f 1 / g (apply L’H o ˆ pital) f \cdot g = \frac{f}{1/g} \text{ (apply L'Hôpital)} f ⋅ g = 1/ g f (apply L’H o ˆ pital) Convert ∞ − ∞
f − g = 1 / g − 1 / f 1 / ( f g ) (find common denominator) f - g = \frac{1/g - 1/f}{1/(fg)} \text{ (find common denominator)} f − g = 1/ ( f g ) 1/ g − 1/ f (find common denominator) Indeterminate Forms List of Indeterminate Forms
0 0 , ∞ ∞ , 0 ⋅ ∞ , ∞ − ∞ , 0 0 , 1 ∞ , ∞ 0 \tfrac{0}{0},\ \tfrac{\infty}{\infty},\ 0 \cdot \infty,\ \infty - \infty,\ 0^0,\ 1^\infty,\ \infty^0 0 0 , ∞ ∞ , 0 ⋅ ∞ , ∞ − ∞ , 0 0 , 1 ∞ , ∞ 0 Exponential Indeterminate (general)
y = f ( x ) g ( x ) ⟹ ln y = g ( x ) ln f ( x ) y = f(x)^{g(x)} \implies \ln y = g(x) \ln f(x) y = f ( x ) g ( x ) ⟹ ln y = g ( x ) ln f ( x ) Limits at Infinity Rational Function (deg numerator < denominator)
lim x → ∞ P ( x ) Q ( x ) = 0 \lim_{x \to \infty} \frac{P(x)}{Q(x)} = 0 x → ∞ lim Q ( x ) P ( x ) = 0 Rational Function (deg numerator = denominator)
lim x → ∞ a n x n + ⋯ b n x n + ⋯ = a n b n \lim_{x \to \infty} \frac{a_n x^n + \cdots}{b_n x^n + \cdots} = \frac{a_n}{b_n} x → ∞ lim b n x n + ⋯ a n x n + ⋯ = b n a n Rational Function (deg numerator > denominator)
lim x → ∞ P ( x ) Q ( x ) = ± ∞ \lim_{x \to \infty} \frac{P(x)}{Q(x)} = \pm\infty x → ∞ lim Q ( x ) P ( x ) = ± ∞ Exponential Beats Polynomial
lim x → ∞ x n e x = 0 \lim_{x \to \infty} \frac{x^n}{e^x} = 0 x → ∞ lim e x x n = 0 Polynomial Beats Logarithm
lim x → ∞ ln x x n = 0 ( n > 0 ) \lim_{x \to \infty} \frac{\ln x}{x^n} = 0 \quad(n > 0) x → ∞ lim x n ln x = 0 ( n > 0 ) One-Sided Limits & Continuity Left-Hand Limit
lim x → a − f ( x ) = L \lim_{x \to a^-} f(x) = L x → a − lim f ( x ) = L Right-Hand Limit
lim x → a + f ( x ) = L \lim_{x \to a^+} f(x) = L x → a + lim f ( x ) = L Two-Sided Limit Exists
lim x → a f ( x ) = L ⟺ lim x → a − f = lim x → a + f = L \lim_{x \to a} f(x) = L \iff \lim_{x \to a^-} f = \lim_{x \to a^+} f = L x → a lim f ( x ) = L ⟺ x → a − lim f = x → a + lim f = L Continuity at a
f continuous at a ⟺ lim x → a f ( x ) = f ( a ) f \text{ continuous at } a \iff \lim_{x \to a} f(x) = f(a) f continuous at a ⟺ x → a lim f ( x ) = f ( a ) Squeeze Theorem
g ( x ) ≤ f ( x ) ≤ h ( x ) , lim g = lim h = L ⟹ lim f = L g(x) \le f(x) \le h(x),\ \lim g = \lim h = L \implies \lim f = L g ( x ) ≤ f ( x ) ≤ h ( x ) , lim g = lim h = L ⟹ lim f = L 