Limit, Continuity and Differentiability Cheatsheet

Cheat Codes & Shortcuts

• Limit Definition: \( \lim_{x \to a} f(x) = L \) if for every \( \epsilon > 0 \), there exists \( \delta > 0 \) such that \( |f(x) - L| < \epsilon \) when \( 0 < |x - a| < \delta \).
• Continuity: \( f(x) \) is continuous at \( x = a \) if \( \lim_{x \to a} f(x) = f(a) \).
• Differentiability: \( f(x) \) is differentiable at \( x = a \) if \( f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \) exists.
• Limit Rules: \( \lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x) \).
• L’Hôpital’s Rule: If \( \lim_{x \to a} \frac{f(x)}{g(x)} \) is \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), then \( \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} \).
• One-Sided Limits: \( \lim_{x \to a^-} f(x) \) (left) and \( \lim_{x \to a^+} f(x) \) (right).
• Derivative Rules: Product: \( (fg)' = f'g + fg' \); Quotient: \( \left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2} \).
• Chain Rule: If \( y = f(g(x)) \), then \( \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \).
• Implicit Differentiation: Differentiate both sides with respect to \( x \), solve for \( \frac{dy}{dx} \).
• Continuity Implies: Intermediate Value Theorem and Extreme Value Theorem apply.

Quick Reference Table

Type	Form	Solution/Method
Limit	\( \lim_{x \to 2} \frac{x^2 - 4}{x - 2} \)	Simplify: \( \lim_{x \to 2} (x + 2) = 4 \)
L’Hôpital’s	\( \lim_{x \to 0} \frac{\sin x}{x} \)	Apply L’Hôpital: \( \lim_{x \to 0} \frac{\cos x}{1} = 1 \)
Continuity	\( f(x) = \frac{x^2 - 1}{x - 1} \) at \( x = 1 \)	Check if \( \lim_{x \to 1} f(x) = f(1) \); undefined at \( x = 1 \)
Derivative	\( f(x) = x^2 \sin x \)	Product Rule: \( f'(x) = 2x \sin x + x^2 \cos x \)
Chain Rule	\( f(x) = \sin(x^2) \)	\( f'(x) = \cos(x^2) \cdot 2x \)
Implicit	\( x^2 + y^2 = 1 \)	Differentiate: \( 2x + 2y \frac{dy}{dx} = 0 \), solve for \( \frac{dy}{dx} \)

Advice

Limit, Continuity and Differentiability Quick Tips

• Limits: Try direct substitution; if indeterminate, simplify or use L’Hôpital’s Rule.
• Continuity: Check \( \lim_{x \to a} f(x) = f(a) \) and that \( f(a) \) is defined.
• Differentiability: Requires continuity; compute \( \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \).
• Chain Rule: Apply when function is composite: \( (f \circ g)'(x) = f'(g(x)) g'(x) \).
• Implicit Differentiation: Useful for equations not solved for \( y \).

Limit, Continuity and Differentiability Speed Quiz

Test your speed with 5 limit, continuity, and differentiability questions! You have 30 seconds per question.