Answer:
[tex]\displaystyle \lim_{x \to 0} \frac{1 - \cos x}{x} = 0[/tex]
General Formulas and Concepts:
Calculus
Limits
- Limit Rule [Variable Direct Substitution]: [tex]\displaystyle \lim_{x \to c} x = c[/tex]
Special Limit Rule [L’Hopital’s Rule]: [tex]\displaystyle \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}[/tex]
Differentiation
- Derivatives
- Derivative Notation
Derivative Property [Addition/Subtraction]: [tex]\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)][/tex]
Basic Power Rule:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
Step-by-step explanation:
Step 1: Define
Identify
[tex]\displaystyle \lim_{x \to 0} \frac{1 - \cos x}{x}[/tex]
Step 2: Find
- Special Limit Rule [L'Hopital's Rule]: [tex]\displaystyle \lim_{x \to 0} \frac{1 - \cos x}{x} = \lim_{x \to 0} \frac{(1 - \cos x)'}{(x)'}[/tex]
- Rewrite [Derivative Rule - Addition/Subtraction]: [tex]\displaystyle \lim_{x \to 0} \frac{1 - \cos x}{x} = \lim_{x \to 0} \frac{(1)' - (\cos x)'}{(x)'}[/tex]
- Basic Power Rule: [tex]\displaystyle \lim_{x \to 0} \frac{1 - \cos x}{x} = \lim_{x \to 0} \frac{0 - (\cos x)'}{1}[/tex]
- Simplify: [tex]\displaystyle \lim_{x \to 0} \frac{1 - \cos x}{x} = \lim_{x \to 0} -(\cos x)'[/tex]
- Trigonometric Differentiation: [tex]\displaystyle \lim_{x \to 0} \frac{1 - \cos x}{x} = \lim_{x \to 0} \sin x[/tex]
- Evaluate Limit [Limit Rule - Variable Direct Substitution]: [tex]\displaystyle \lim_{x \to 0} \frac{1 - \cos x}{x} = 0[/tex]
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Advanced Limit Techniques
