Limits and Derivatives

• Introduction to Limits

  • Intuitive idea of a limit
  • Left-hand limit (LHL): $\lim_{x \to a^-} f(x)$
  • Right-hand limit (RHL): $\lim_{x \to a^+} f(x)$
  • Existence of Limit: $\text{LHL} = \text{RHL} = \lim_{x \to a} f(x)$

• Algebra of Limits

  • Sum Rule: $\lim [f(x) + g(x)] = \lim f(x) + \lim g(x)$
  • Difference Rule: $\lim [f(x) - g(x)] = \lim f(x) - \lim g(x)$
  • Product Rule: $\lim [f(x) \cdot g(x)] = \lim f(x) \cdot \lim g(x)$
  • Quotient Rule: $\lim \frac{f(x)}{g(x)} = \frac{\lim f(x)}{\lim g(x)}$, where $\lim g(x) \neq 0$

• Limits of Standard Functions

  • Polynomial and Rational Functions
  • Trigonometric Limits
    • $\lim_{x \to 0} \frac{\sin x}{x} = 1$
    • $\lim_{x \to 0} \frac{1 - \cos x}{x} = 0$
  • Exponential and Logarithmic Limits
    • $\lim_{x \to 0} \frac{e^x - 1}{x} = 1$
    • $\lim_{x \to 0} \frac{\log_e(1 + x)}{x} = 1$

• Standard Algebraic Limits (Formulae)

  • $\lim_{x \to a} \frac{x^n - a^n}{x - a} = n a^{n-1}$

• Derivatives

  • Geometric Meaning: Slope of the tangent line
  • Physical Meaning: Rate of change
  • First Principle of Derivative: $f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$

• Algebra of Derivatives

  • Derivative of Sum: $\frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x)$
  • Derivative of Difference: $\frac{d}{dx}[f(x) - g(x)] = f'(x) - g'(x)$
  • Product Rule: $\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$
  • Quotient Rule: $\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$

• Standard Derivatives (Formulae)

  • $\frac{d}{dx}(x^n) = n x^{n-1}$
  • $\frac{d}{dx}(\sin x) = \cos x$
  • $\frac{d}{dx}(\cos x) = -\sin x$
  • $\frac{d}{dx}(\tan x) = \sec^2 x$
  • $\frac{d}{dx}(e^x) = e^x$
  • $\frac{d}{dx}(\log_e x) = \frac{1}{x}$