Engineering Handbook/Calculus/Limit

Definitions
To say that


 * $$ \lim_{x \to p}f(x) = L, \, $$

means that ƒ(x) can be made as close as desired to L by making x close enough, but not equal, to p.

The following definitions (known as (ε, δ)-definitions) are the generally accepted ones for the limit of a function in various contexts.

Limits for general functions

 * $$\text{If }\lim_{x \to c} f(x) = L_1 \text{ and }\lim_{x \to c} g(x) = L_2 \text{ then:}$$


 * $$\lim_{x \to c} \, [f(x) \pm g(x)] = L_1 \pm L_2$$


 * $$\lim_{x \to c} \, [f(x)g(x)] = L_1 \times L_2$$


 * $$\lim_{x \to c} \frac{f(x)}{g(x)} = \frac{L_1}{L_2} \qquad \text{ if } L_2 \ne 0$$


 * $$\lim_{x \to c} \, f(x)^n = L_1^n \qquad \text{ if }n \text{ is a positive integer}$$


 * $$\lim_{x \to c} \, f(x)^{1 \over n} = L_1^{1 \over n} \qquad \text{ if }n \text{ is a positive integer, and if } n \text{ is even, then } L_1 > 0$$


 * $$\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)} \qquad \text{ if } \lim_{x \to c} f(x) = \lim_{x \to c} g(x) = 0 \text{ or } \lim_{x \to c} |g(x)| = +\infty$$ (L'Hôpital's rule)

Limits of general functions

 * $$\lim_{h\to 0}{f(x+h)-f(x)\over h}=f'(x)$$


 * $$\lim_{h\to0}\left(\frac{f(x+h)}{f(x)}\right)^\frac{1}{h}=\exp\left(\frac{f'(x)}{f(x)}\right)$$


 * $$\lim_{h \to 0}{ \left({f(x(1+h))\over{f(x)}}\right)^{1\over{h}} }=\exp\left(\frac{x f'(x)}{f(x)}\right)$$

Notable special limits

 * $$\lim_{x\to+\infty} \left(1+\frac{k}{x}\right)^{mx}=e^{mk}$$


 * $$\lim_{x\to+\infty} \left(1-\frac{1}{x}\right)^x=\frac{1}{e}$$


 * $$\lim_{x\to+\infty} \left(1+\frac{k}{x}\right)^x=e^k$$


 * $$\lim_{n\to\infty} \frac{n}{\sqrt[n]{n!}}=e$$


 * $$\lim_{n\to \infty }\, 2^{n} \underbrace{\sqrt{2-\sqrt{2+\sqrt{2+\text{...} +\sqrt{2}}}}}_n= \pi$$

Simple functions

 * $$\lim_{x \to c} a = a$$


 * $$\lim_{x \to c} x = c$$


 * $$\lim_{x \to c} ax + b = ac + b$$


 * $$\lim_{x \to c} x^r = c^r \qquad \mbox{ if } r \mbox{ is a positive integer}$$


 * $$\lim_{x \to 0^+} \frac{1}{x^r} = +\infty$$


 * $$\lim_{x \to 0^-} \frac{1}{x^r} = \begin{cases} -\infty, & \text{if } r \text{ is odd} \\ +\infty, & \text{if } r \text{ is even}\end{cases} $$

Logarithmic and exponential functions

 * $$\mbox{For } a > 1: \,$$


 * $$\lim_{x \to 0^+} \log_a x = -\infty$$


 * $$\lim_{x \to \infty} \log_a x = \infty$$


 * $$\lim_{x \to -\infty} a^x = 0$$


 * $$\mbox{If } a < 1: \,$$


 * $$\lim_{x \to -\infty} a^x = \infty$$

Trigonometric functions

 * $$\lim_{x \to a} \sin x = \sin a$$


 * $$\lim_{x \to a} \cos x = \cos a$$


 * $$\lim_{x \to 0} \frac{\sin x}{x} = 1$$


 * $$\lim_{x \to 0} \frac{1-\cos x}{x} = 0$$


 * $$\lim_{x \to 0} \frac{1-\cos x}{x^2} = \frac{1}{2}$$


 * $$\lim_{x \to n^\pm} \tan \left(\pi x + \frac{\pi}{2}\right) = \mp\infty \qquad \text{for any integer } n$$

Near infinities

 * $$\lim_{x\to\infty}N/x=0 \text{ for any real }N $$
 * $$\lim_{x\to\infty}x/N=\begin{cases} \infty, & N > 0 \\ \text{does not exist}, & N = 0 \\ -\infty, & N < 0 \end{cases}$$
 * $$\lim_{x\to\infty}x^N=\begin{cases} \infty, & N > 0 \\ 1, & N = 0 \\ 0, & N < 0 \end{cases}$$
 * $$\lim_{x\to\infty}N^x=\begin{cases} \infty, & N > 1 \\ 1, & N = 1 \\ 0, & 0 < N < 1 \end{cases}$$
 * $$\lim_{x\to\infty}N^{-x}=\lim_{x\to\infty}1/N^{x}=0 \text{ for any } N > 1$$
 * $$\lim_{x\to\infty}\sqrt[x]{N}=\begin{cases} 1, & N > 0 \\ 0, & N = 0 \\ \text{does not exist}, & N < 0 \end{cases}$$
 * $$\lim_{x\to\infty}\sqrt[N]{x}= \infty \text{ for any } N > 0 $$
 * $$\lim_{x\to\infty}\log x=\infty$$
 * $$\lim_{x\to0^+}\log x=-\infty$$