Limits
Limits and continuity
Let $f:I \to \mathbb{R}$ be a function defined on some open interval $I$
of $\mathbb{R}$ except possibly at the point $a$.
$f(x)$ tends to $l$ as $x$ tends to $a$,
$\lim\limits_{x \to a} f(x) = l$, if for every sequence $x_n$ in $I$
with $x_n \to a$ and $x_n \neq a$ for all $n$ the sequence $f(x_n)$
converges to $l$.
$f(x)$ tends to $l$ as $x$ tends to $a$ from the left, $\lim\limits_{x \to a-} f(x) = l$, if for every sequence $x_n$ in $I$ with $x_n \to a$ and $x_n < a$ for all $n$ we have $f(x_n) \to l$.
$f(x)$ tends to $l$ as $x$ tends to $a$ from the right, $\lim\limits_{x \to a+}f(x)=l$, if for every sequence $x_n$ in $I$ with $x_n \to a$ and $x_n > a$ for all $n$ we have $f(x_n) \to l$.
$\lim\limits_{x \to a}f(x)$ exists and equals $l$ if and only if $\lim\limits_{x \to a-}f(x)$ and $\lim\limits_{x \to a+}f(x)$ both exist and equal $l$.
Limits of floor and ceiling functions
For any integer $k$:
| $\lim\limits_{x \to k-}\lfloor x\rfloor=k-1$ | $\lim\limits_{x \to k+}\lfloor x\rfloor = k$ |
| $\lim\limits_{x\to k-} \lceil x \rceil = k$ | $\lim\limits_{x \to k+} \lceil x \rceil = k+1$ |
Combination rules for limits
If $\lim\limits_{x \to a} f(x) = l$ and $\lim\limits_{x \to a}g(x)=m$ then
| sum rule | $\lim\limits_{x \to a}(f(x)+g(x)) = l+m$ |
| multiple rule | $\lim\limits_{x \to a} \lambda f(x) = \lambda l$, $\lambda \in \mathbb{R}$ |
| product rule | $\lim\limits_{x \to a} f(x)g(x) = lm$ |
| quotient rule | $\lim\limits_{x \to a} f(x)/g(x) = l/m$, $m \neq 0$ |
Squeeze rule for limits: If $f(x) \leq g(x) \leq h(x)$ for $x \neq a$, $\lim\limits_{x \to a} f(x) = l$ and $\lim\limits_{x \to a} h(x) = l$ then $\lim\limits_{x \to a} g(x) = l$.
Continuity
Let $f:D \to \mathbb{R}$ be a function defined on some subset $D$ of $\mathbb{R}$. We say that $f$ is continuous at a point $a \in D$ if $\lim\limits_{x \to a} f(x)$ exists and equals $f(a)$. We say that $f$ is continuous if it is continuous at $a$ for every $a \in D$.
Combination rules for continuous functions
If $f$ and $g$ are continuous at $a$, then so are
| the sum | $f+g$ |
| the multiple | $\lambda f$, $\lambda \in \mathbb{R}$ |
| the product | $fg$ |
| the quotient | $f/g$, $g(a) \neq 0$. |
If $f$ if continuous at $a$ and $g$ is continuous at $f(x)$ then the composite function $g \circ f$ is continuous at $a$.
The value of $\lim\limits_{x \to a} f(x)$ does not depend on the value of $f(a)$ at $a$. The limit can exist even when $f$ is not defined at $a$.