Mathematics · Ch 7 — Limits and Derivatives

Summary

Limits. $\displaystyle\lim_{x\to a}f(x)=L$ exists iff the

left-hand and right-hand limits both equal $L$ . Algebra of limits: limits of a

sum/difference/product/quotient (denominator limit $\neq0$ ) equal the

sum/difference/product/quotient of the individual limits.

Standard limit results.

\lim_{x\to a}\frac{x^n-a^n}{x-a}=na^{n-1},\qquad \lim_{x\to0}\frac{\sin x}{x}=1,\qquad \lim_{x\to0}\frac{\tan x}{x}=1,

\lim_{x\to0}\frac{e^x-1}{x}=1,\qquad \lim_{x\to0}\frac{\ln(1+x)}{x}=1,\qquad \lim_{x\to0}\frac{a^x-1}{x}=\ln a.

Polynomials, rational functions (denominator nonzero), $\sin x$ , $\cos x$ , $e^x$ and $\ln x$

(where defined) are all evaluated by direct substitution; a $0/0$ form is resolved by

factoring (polynomials/rationals) or by scaling to a standard limit

(trigonometric/exponential/logarithmic).

Derivative -- two equivalent meanings, one limit.

f'(x) = \lim_{h\to0}\frac{f(x+h)-f(x)}{h}

is simultaneously the instantaneous rate of change of $y=f(x)$ (generalising average

velocity $\Delta s/\Delta t$ ) and the slope of the tangent to $y=f(x)$ at that point (the

limiting slope of a secant line as the second point slides in).

Algebra of derivatives, derived directly from this limit definition:

(u\pm v)' = u'\pm v', \qquad (uv)' = u'v+uv', \qquad \left(\frac{u}{v}\right)' = \frac{u'v-uv'}{v^2}\ (v\neq0).

Derivatives of polynomial and trigonometric functions. …