Limits & continuity formulas

Use one-sided limits for $|x|$, floor, piecewise functions and any function that changes formula at $a$.

Only for $x$ in radians and $x\to 0$. For $x\to \infty$, $\sin x/x\to 0$ instead.

Each is a $0/0$ form that equals a derivative at $0$ (of $e^x$, $a^x$ and $\ln (1+x)$ respectively).

Here $n$ is any real number. More generally ${\lim }_{x\to \infty }(1+k/x{)}^x=e^k$ for any real $k$.

Applies when $f(x)>0$ near $a$, $f(x)\to 1$, $g(x)\to \pm \infty$, and the limit inside the exponential exists.

The $e^x$ series holds for all real $x$; the $\ln (1+x)$ series holds for $-1<x\le 1$.

The $\sin x$ and $\cos x$ series hold for all real $x$; the $\tan x$ series for $|x|<\pi /2$. In a $0/0$ limit at $0$, replace each function by enough terms and compare the lowest powers: for example $\frac{x-\sin x}{x^3}=\frac{x^3/6-\cdots }{x^3}\to \frac{1}{6}$.

Needs: $f/g$ of the form $0/0$ or $\pm \infty /\pm \infty$ as $x\to a$; $f,g$ differentiable with $g^{\prime }(x)\ne 0$ on a punctured neighbourhood of $a$; and the limit of $f^{\prime }/g^{\prime }$ (the limit on the right of the equation) to exist, finite or infinite. It also holds for one-sided limits and for $a=\pm \infty$. Differentiate numerator and denominator separately; do not use the quotient rule.

Exponentials beat powers and powers beat logarithms as $x\to \infty$.

Three things must hold: $f(a)$ is defined, the limit exists and is finite, and the two are equal. Equivalently, left limit = right limit = $f(a)$.

Used to show a root exists, for example $f(a)f(b)<0$ gives a root of $f$ in $(a,b)$.

Both conditions matter: on the open interval $(0,1)$, $f(x)=1/x$ is continuous but has no maximum.

$f$ is differentiable at $a$ when this limit exists and is finite.

$f$ is differentiable at $a$ if and only if both exist, are finite and $f_{-}^{\prime }(a)=f_{+}^{\prime }(a)$.

The converse is false. Hence a function that is not continuous at $a$ cannot be differentiable at $a$.

For $u,v$ differentiable at the point.

For $g$ differentiable at $x$ and $f$ differentiable at $g(x)$.

For $u>0$ and $u,v$ differentiable. For example $\frac{d}{dx}{x}^x=x^x(1+\ln x)$ for $x>0$.

All three conditions are needed. Typical use: if $f$ is continuous on the closed interval between two roots and differentiable inside it, then $f^{\prime }$ has a root strictly between them.

Some point has tangent slope equal to the average slope over $[a,b]$. Rolle's theorem is the special case $f(a)=f(b)$.

The condition $g^{\prime }\ne 0$ ensures $g(b)\ne g(a)$ (by Rolle's theorem). Taking $g(x)=x$ gives Lagrange's theorem.