Functions and limits

Calculus is about how functions change, so we first need a reliable stock of functions and a working notion of limit.

1. The standard functions: exponentials and logarithms (Def. 1.2), inverse trig (Def. 1.3) and hyperbolic functions (Def. 1.4, Prop. 1.5).
2. Limits (Def. 1.6): what $f(x)$ approaches, computed with the limit laws (Technique 1.7) and a few standard limits (Prop. 1.8).
3. Continuity (Def. 1.9) and the behaviour at infinity (Def. 1.10). The rigorous $\varepsilon$–$\delta$ theory is in Analysis I; here the aim is fluent computation.

A real function $f$ assigns to each $x$ in its domain $D\subseteq\mathbb{R}$ one value $f(x)$; its range is $\{f(x): x\in D\}$. The natural domain of a formula is the largest set where it makes sense (no division by $0$, square roots of negatives, logs of non-positives). The composite $(f\circ g)(x) = f(g(x))$ is defined where $g(x)$ lies in the domain of $f$. If $f$ is one-to-one (strictly increasing or decreasing on its domain), it has an inverse $f^{-1}$ with $f^{-1}(f(x)) = x$; its graph is the reflection of $f$'s in $y = x$. $f$ is even if $f(-x) = f(x)$ and odd if $f(-x) = -f(x)$.

$e^x$ is the exponential function ($e\approx2.71828$); it is positive, strictly increasing, and $e^{a+b} = e^ae^b$, $(e^a)^b = e^{ab}$. Its inverse is the natural logarithm $\ln: (0, \infty)\to\mathbb{R}$, so $e^{\ln x} = x$ and $\ln e^x = x$. Laws: $\ln(ab) = \ln a + \ln b$, $\ln\frac ab = \ln a - \ln b$, $\ln a^r = r\ln a$. General powers: $a^x = e^{x\ln a}$ for $a > 0$, and $\log_ax = \frac{\ln x}{\ln a}$.

Restrict the trig functions to intervals where they are one-to-one, and invert:

• $\arcsin: [-1, 1]\to[-\frac\pi2, \frac\pi2]$, the inverse of $\sin$ on $[-\frac\pi2, \frac\pi2]$;
• $\arccos: [-1, 1]\to[0, \pi]$, the inverse of $\cos$ on $[0, \pi]$;
• $\arctan: \mathbb{R}\to(-\frac\pi2, \frac\pi2)$, the inverse of $\tan$ on $(-\frac\pi2, \frac\pi2)$. So $\sin(\arcsin x) = x$ for $x\in[-1, 1]$, but $\arcsin(\sin x) = x$ only for $x\in[-\frac\pi2, \frac\pi2]$.

$\cosh$ is even with minimum $\cosh0 = 1$; $\sinh$ is odd and strictly increasing; $\tanh$ is odd with range $(-1, 1)$. (A hanging chain has the shape of a $\cosh$ curve, the catenary.)

(i) $\cosh^2x - \sinh^2x = 1$; $\sinh2x = 2\sinh x\cosh x$; $\cosh2x = \cosh^2x + \sinh^2x$. (ii) Osborn's rule: a trig identity becomes a hyperbolic one by replacing $\cos\to\cosh$, $\sin\to\sinh$ and changing the sign of each product of two sines. (iii) $\operatorname{arsinh}x = \ln\big(x + \sqrt{x^2 + 1}\big)$ for all $x$; $\operatorname{arcosh}x = \ln\big(x + \sqrt{x^2 - 1}\big)$ for $x\ge1$; $\operatorname{artanh}x = \frac12\ln\frac{1 + x}{1 - x}$ for $|x| < 1$.

$\lim_{x\to a}f(x) = L$ means $f(x)$ can be made as close to $L$ as we like by taking $x$ close enough to $a$ (but $x\ne a$). Precisely: for every $\varepsilon > 0$ there is $\delta > 0$ with $|f(x) - L| < \varepsilon$ whenever $0 < |x - a| < \delta$. One-sided limits $x\to a^-$, $x\to a^+$ approach from one side; the limit exists iff both exist and agree. The value $f(a)$ itself plays no role.

Limit laws: limits of sums, products, quotients (non-zero denominator limit) and compositions with continuous functions are what you expect. If direct substitution gives a number, that's the limit. For $\frac00$ forms: (1) factorise and cancel; (2) rationalise (multiply by a conjugate) for square roots; (3) use a standard limit (see Prop. 1.8, next); (4) squeeze: if $g\le f\le h$ near $a$ and $g, h\to L$ then $f\to L$; (5) later, L'Hôpital or Taylor series.

Also, as $x\to\infty$: $\frac{x^k}{e^x}\to0$ and $\frac{\ln x}{x^k}\to0$ for every $k > 0$ (exponentials beat powers, powers beat logs).

$f$ is continuous at $a$ if $\lim_{x\to a}f(x) = f(a)$, and continuous on an interval if continuous at every point. Polynomials, rational functions (where defined), $e^x$, $\ln$, trig and their combinations are continuous. Intermediate value theorem (proved in Analysis I): if $f$ is continuous on $[a, b]$ and $c$ is between $f(a)$ and $f(b)$, then $f(x) = c$ for some $x\in[a, b]$. In particular a sign change means a root.

$\lim_{x\to\infty}f(x) = L$ means $f(x)$ gets close to $L$ for large $x$; then $y = L$ is a horizontal asymptote. If $f(x)\to\pm\infty$ as $x\to a$, $x = a$ is a vertical asymptote. If $f(x) - (mx + c)\to0$, $y = mx + c$ is an oblique asymptote. For rational functions, divide by the highest power of $x$ in the denominator; polynomial division gives oblique asymptotes.

1. Substitute. A number? Done. Non-zero over zero? Infinite limit (check signs on each side).
2. $\frac00$: factorise/cancel, rationalise, or reduce to a standard limit.
3. $x\to\infty$: divide by the dominant term; remember $\ln x\ll x^k\ll e^x$.
4. Oscillating factors ($\sin\frac1x$): squeeze.
5. Check numerically with a value close to $a$ if unsure.