Taylor equivalents as ${u\to0}$

Piece	Replace by
${e^u-1}$	$u$
${\ln(1+u)}$	$u$
$\sin u$, $\tan u$, $\arctan u$	$u$
${1-\cos u}$	$\tfrac{u^2}2$
${\sqrt{1+u}-1}$	$\tfrac u2$
${\sqrt[3]{1+u}-1}$	$\tfrac u3$
${(1+u)^a-1}$	$a\,u$
${a^u-1}$	$u\ln a$
${u^2+u}$, sum with ${u\to0}$	$u$ (lowest power)
${u^2+u}$, sum with ${u\to\infty}$	$u^2$ (highest power)

$u$ can be $x$, $x^2$, ${-x^2}$ or $\sin x$. For ${x\to+\infty}$ substitute ${u=\tfrac1x}$.

Scale of infinities as ${n\to+\infty}$

$\ln n\ \ll\ n^{a}\ {(a\gt0)}\ \ll\ k^{n}\ {(k\gt1)}\ \ll\ n!\ \ll\ n^{n}$

Fraction of sums: keep the biggest term on top and on the bottom, then cancel.
Negative exponents are small, not big: ${n^{-3}\to0}$, ${e^{-n}\to0}$, ${3^{-n}\to0}$.
Bounded over infinite tends to $0$: $\dfrac{\sin n}{n}$, ${\dfrac{(-1)^n}{n}}$, ${\dfrac{\cos n}{n+1}}$.
A numerator tending to ${+\infty}$ over a denominator tending to ${0^+}$ gives ${+\infty}$.
${a_n=o(b_n)}$ means ${a_n/b_n\to0}$. ${\ a_n\sim b_n}$ means ${a_n/b_n\to1}$.
$k^n$ tends to ${+\infty}$ if ${k\gt1}$, equals $1$ if ${k=1}$, tends to $0$ if ${|k|\lt1}$, and is irregular if ${k\le-1}$.

Series

Series	Behaviour
${\sum_{n=0}^{+\infty}q^n}$, ${{\|q\|\lt1}}$	${=\dfrac1{1-q}}$; from ${n=1}$: ${\dfrac q{1-q}}$
$\sum q^n$, ${q\ge1}$	diverges to ${+\infty}$ (${q\le-1}$: irregular)
$\sum\dfrac1{n^p}$	converges iff ${p\gt1}$
$\sum\dfrac1n$, $\sum\dfrac1{\sqrt n}$, $\sum\dfrac1{\ln n}$	diverge to ${+\infty}$
$\sum_{n=1}^{+\infty}\dfrac1{n(n+1)}$	${=1}$ (telescoping)
$\sum\dfrac{n^k}{e^n}$, $\sum\dfrac{n^k}{2^n}$	converge (compare with $\tfrac1{n^2}$)
${\sum\dfrac{(-1)^n}{n^p}}$, ${{p\gt1}}$	absolutely convergent
${\sum\dfrac{(-1)^n}{n}}$	convergent, not absolutely
term does not tend to $0$	does not converge (necessary condition)

For a fraction of polynomials in $n$ the term behaves like $\dfrac{c}{n^{p}}$ with $p=(\deg\text{ bottom})-(\deg\text{ top})$, and the series converges iff ${p\gt1}$.

Two-variable limits at $(0,0)$

Set ${y=0}$, simplify, read the number.
Set ${x=0}$, simplify, read the number.
Different numbers: does not exist. Same number: try ${y=x}$ as well.
If the denominator is ${x^2+y^2}$ and every term on top has degree $3$ or more, the limit is $0$.

The past ratios ${\dfrac{2x^2}{x^2+y^2}}$, ${\dfrac{2x^2+y^2}{x^2+y^2}}$ and ${\dfrac{x+y}{x-y^2}}$ do not have a limit: two paths (${y=0}$, ${x=0}$ or ${y=x}$) give different values. A higher degree on top, as in ${\dfrac{x^2y}{x^2+y^2}}$, gives $0$.

Counterexamples that kill options

Claim to break	Counterexample
closed ⇒ every point is a boundary point	$[0,1]$: $\tfrac12$ is an interior point
closed ⇒ every point is a limit point	$\{0\}$: $0$ is an isolated point
open ⇒ not closed	$\R^n$ and $\varnothing$ are both
all points are boundary points ⇒ closed	$\mathbb{Q}$
an infinite intersection of open sets is open	$\bigcap_{n=1}^{+\infty}\left(-\tfrac1n,\tfrac1n\right)=\{0\}$
bounded above ⇒ has a max; the sup is a limit point; the sup is isolated	$[0,1)$: sup $1$, no max, not isolated. ${[0,1]\cup\{2\}}$: sup $2$, not a limit point
quasi-concave ⇒ concave; coercive ⇒ supercoercive	${e^{-x^2}}$
convex lower contour sets ⇒ convex function	$x^3$
local maximizer ⇒ global maximizer	${x^3-x}$, ${x+\tfrac1x}$
concave ⇒ has a max	$\ln x$, $x$
bounded sequence ⇒ convergent	${(-1)^n}$, $\sin n$
unbounded ⇒ divergent to ${+\infty}$	${(-1)^nn}$
divergent to ${+\infty}$ ⇒ increasing	$n^2$ for $n$ odd, $n$ for $n$ even
increasing ⇒ divergent	${1-\tfrac1n}$
convergent series ⇒ absolutely convergent; ${a_n\to0}$ ⇒ the series converges	${\sum\tfrac{(-1)^n}n}$; $\sum\tfrac1n$
continuous on $\R$ ⇒ bounded, has a zero, monotone, has a max	$x$, $1$, $x^2$, $e^x$
the sum theorem for limits applies	${x^2+\sin x}$ as ${x\to+\infty}$ ($\sin x$ has no limit); ${x^2-x}$ as ${x\to+\infty}$ and ${e^{1/x}-\tfrac1x}$ as ${x\to0^+}$ (${+\infty-\infty}$)

Quantifiers the graders check

Definition	Must contain
convex set, concave, quasi-concave	${\forall\vx,\vy\in C}$, ${\forall\alpha\in[0,1]}$, and the point ${\alpha\vx+(1-\alpha)\vy}$
${a_n\to l}$	$\forall\varepsilon\gt0\ \exists\bar n\in\N$: ${\forall n\ge\bar n}$, $l-\varepsilon\lt a_n\lt l+\varepsilon$
${a_n\to+\infty}$	${\forall M\gt0\ \exists\bar n}$: ${\forall n\ge\bar n}$, ${a_n\gt M}$
${\lim_{x\to c}f(x)=l}$	$c$ a limit point of $A$; $\forall\varepsilon\gt0\ \exists\delta\gt0$: ${x\in A}$, ${c-\delta\lt x\lt c+\delta}$, ${x\ne c}$ ${\Rightarrow}$ $l-\varepsilon\lt f(x)\lt l+\varepsilon$
continuity at $c$	both cases: limit point (${\lim=f(c)}$) and isolated point (always continuous)
$\sup A$	an upper bound and ${\le}$ every upper bound
coercive on $C$	$\exists t$: ${(f\ge t)\cap C}$ compact and non-empty
supercoercive on $\R^n$	$\forall$ sequence ${\vx_n\in\R^n}$ such that ${\\|\vx_n\\|\to+\infty}$, ${f(\vx_n)\to-\infty}$
series	define $s_n$ first; the series is (the limit of) $s_n$

Proofs asked in past papers

Proof	Times	Card
Limit of a sum of functions, finite limits	3	T15
Fenchel's theorem, concave case	2	T7
Regularity of a series with terms ${a_n\ge0}$	2	T11
Triangle inequality, and the Pythagorean theorem in $\R^n$	1 each	T1, T2
Tonelli's theorem	1	D11
Convergent ⇒ bounded	1	T9
Increasing and unbounded ⇒ divergent to ${+\infty}$	1	T10
Concave ⇒ quasi-concave	1	T5
Convex ⇒ convex lower contour sets	1	T6
Uniqueness of the max (and of the sup)	1	D3
$A$ open ${\iff}$ $A^C$ closed	1	T3
Limit of a sum of sequences, finite limits	1	T13
Neighbourhoods are open (mock)	1	T4
$\sum_{n=1}^{+\infty}\frac1{n(n+1)}=1$ by telescoping (mock)	1	T12