Mathematics · Ch 12 — Limits and Derivatives

Summary

Intuitive idea of a limit: $\lim_{x \to a} f(x) = L$ means $f(x)$ gets arbitrarily close to $L$ as $x$ gets arbitrarily close to $a$ (but $x \neq a$ ). The limit may exist even if $f(a)$ is undefined.
Standard limits:
- $\lim_{x \to a} \frac{x^n - a^n}{x - a} = n a^{n-1}$ (for any rational $n$ ).
- $\lim_{x \to 0} \frac{\sin x}{x} = 1$ (with $x$ in radians).
- $\lim_{x \to 0} \frac{1 - \cos x}{x} = 0$ .
Algebra of limits: If $\lim f(x) = L$ $lim f (x) = L$ and $\lim g(x) = M$ $lim g (x) = M$ , then
- $\lim [f(x) \pm g(x)] = L \pm M$
- $\lim [f(x) \cdot g(x)] = L \cdot M$
- $\lim \frac{f(x)}{g(x)} = \frac{L}{M}$ (provided $M \neq 0$ ).
Derivative as a limit: $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$ , provided the limit exists.
Derivative of standard functions:
- $\frac{d}{dx}(x^n) = n x^{n-1}$
- $\frac{d}{dx}(\sin x) = \cos x$ , $\frac{d}{dx}(\cos x) = -\sin x$
- $\frac{d}{dx}(\tan x) = \sec^2 x$
- $\frac{d}{dx}(\sec x) = \sec x \tan x$ , $\frac{d}{dx}(\csc x) = -\csc x \cot x$ , $\frac{d}{dx}(\cot x) = -\csc^2 x$ . …