In real analysis, it is mostly necessary to deal with the limits of functions instead of the limits sequences. Therefore we want to now generalize the concept of limits from sequences to functions.

Let \(D\) be a subset of real numbers \(\mathbb R\), \(x\in D\), and let \(f:D\to\mathbb R\) be a function. We define different notations:

$(1)$

Limit of $f$ at $x$: For every real sequence $(\xi_n)_{n\in\mathbb N}$ with $\xi_n\in D$ for all $n\in\mathbb N$, which is convergent to $x$ we have that $\lim_{n\to\infty}f(\xi_n)=y$. Then this can be notated as $$\lim_{\xi\to x} f(\xi)= y.$$ Equivalently, for every $\epsilon > 0$ there is a $\delta > 0$ such that for every $\xi$ with $\xi\in D$ satisfying $0 < |\xi-x| < \delta,$ it follows $|f(\xi)-y| < \epsilon.$$(2)$

Limit of $f$ at $x$ from above: For every real sequence $(\xi_n)_{n\in\mathbb N}$ with $\xi_n\in D$, $\xi_n > x$ for all $n\in\mathbb N$, which is convergent to $x$, we have that $\lim_{n\to\infty}f(\xi_n)=y$. Then this can be notated as $$\lim_{\xi\searrow x} f(\xi)= y.$$ Equivalently, for every $\epsilon > 0$ there is a $\delta > 0$ such that for every $\xi$ with $\xi\in D$ satisfying $0 < \xi-x < \delta,$ it follows $|f(\xi)-y| < \epsilon.$$(3)$

Limit of $f$ at $x$ from below: For every real sequence $(\xi_n)_{n\in\mathbb N}$ with $\xi_n\in D$, $\xi_n < x$ for all $n\in\mathbb N$, which is convergent to $x$ we have that $\lim_{n\to\infty}f(\xi_n)=y$. Then this can be notated as $$\lim_{\xi\nearrow x} f(\xi)= y.$$ Equivalently, for every $\epsilon > 0$ there is a $\delta > 0$ such that for every $\xi$ with $\xi\in D$ satisfying $0 < x-\xi < \delta,$ it follows $|f(x)-y| < \epsilon.$$(4)$

Limit of $f$ at $+\infty$: For every real sequence $(\xi_n)_{n\in\mathbb N}$ with $\xi_n\in D$ for all $n\in\mathbb N$, which tends to infinity, we have that $\lim_{n\to\infty}f(\xi_n)=y$. Then this can be notated as $$\lim_{\xi\to\infty} f(\xi)= y.$$ Equivalently, for every $\epsilon > 0$ there is an $N\in\mathbb N$ such that for every $\xi$ with $\xi\in D$ satisfying $\xi > N$ it follows $|f(\xi)-y| < \epsilon.$$(5)$

Limit of $f$ at $-\infty$: For every real sequence $(\xi_n)_{n\in\mathbb N}$ with $\xi_n\in D$ for all $n\in\mathbb N$, which tends to minus infinity, we have that $\lim_{n\to\infty}f(\xi_n)=y$. Then this can be notated as $$\lim_{\xi\to-\infty} f(\xi)= y.$$ Equivalently, for every $\epsilon > 0$ there is an $N\in\mathbb N$ such that for every $\xi$ with $\xi\in D$ satisfying $\xi < - N$ it follows $|f(\xi)-y| < \epsilon.$$(6)$

Limit of $f$ at $x$ is $\infty$: For every real sequence $(\xi_n)_{n\in\mathbb N}$ with $\xi_n\in D$ for all $n\in\mathbb N$, which is convergent to $x$, we have that the sequence $(f(\xi_n))_{n\in\mathbb N}$ tends to infinity. Then this can be notated as $$\lim_{\xi\to x} f(\xi)= \infty.$$ Equivalently, for every $M\in\mathbb R$ there is an $\delta > 0$ such that for every $\xi$ with $\xi\in D$ satisfying $0 < |\xi-x| < \delta$ it follows $f(x) > M.$$(7)$

Limit of $f$ at $x$ is $-\infty$: For every real sequence $(\xi_n)_{n\in\mathbb N}$ with $\xi_n\in D$ for all $n\in\mathbb N$, which is convergent to $x$, we have that the sequence $(f(\xi_n))_{n\in\mathbb N}$ tends to minus infinity. Then this can be notated as $$\lim_{\xi\to x} f(\xi)= -\infty.$$ Equivalently, for every $M\in\mathbb R$ there is an $\delta > 0$ such that for every $\xi$ with $\xi\in D$ satisfying $0 < |\xi-x| < \delta$ it follows $f(x) < M.$$(8)$

Limit of $f$, as $x$ approaches $\infty$ is $\infty$: For every real sequence $(\xi_n)_{n\in\mathbb N}$ with $\xi_n\in D$ for all $n\in\mathbb N$, which tends to infinity, we have that the sequence $(f(\xi_n))_{n\in\mathbb N}$ also tends to infinity. Then this can be notated as $$\lim_{\xi\to \infty} f(\xi)= \infty.$$ Equivalently, for every $M\in\mathbb R$ there is an $N\in\mathbb R$ such that for every $\xi$ with $\xi\in D$ satisfying $\xi > N$ it follows $f(x) > M.$$(9)$

Limit of $f$, as $x$ approaches $\infty$ is $-\infty$: For every real sequence $(\xi_n)_{n\in\mathbb N}$ with $\xi_n\in D$ for all $n\in\mathbb N$, which tends to infinity, we have that the sequence $(f(\xi_n))_{n\in\mathbb N}$ tends to minus infinity. Then this can be notated as $$\lim_{\xi\to \infty} f(\xi)= -\infty.$$ Equivalently, for every $M\in\mathbb R$ there is an $N\in\mathbb R$ such that for every $\xi$ with $\xi\in D$ satisfying $\xi > N$ it follows $f(x) < M.$$(10)$

Limit of $f$, as $x$ approaches $-\infty$ is $\infty$: For every real sequence $(\xi_n)_{n\in\mathbb N}$ with $\xi_n\in D$ for all $n\in\mathbb N$, which tends to minus infinity, we have that the sequence $(f(\xi_n))_{n\in\mathbb N}$ tends to infinity. Then this can be notated as $$\lim_{\xi\to -\infty} f(\xi)= \infty.$$ Equivalently, for every $M\in\mathbb R$ there is an $N\in\mathbb R$ such that for every $\xi$ with $\xi\in D$ satisfying $\xi < N$ it follows $f(x) > M.$$(11)$

Limit of $f$, as $x$ approaches $-\infty$ is $-\infty$: For every real sequence $(\xi_n)_{n\in\mathbb N}$ with $\xi_n\in D$ for all $n\in\mathbb N$, which tends to minus infinity, we have that the sequence $(f(\xi_n))_{n\in\mathbb N}$ also tends to minus infinity. Then this can be notated as $$\lim_{\xi\to -\infty} f(\xi)= -\infty.$$ Equivalently, for every $M\in\mathbb R$ there is an $N\in\mathbb R$ such that for every $\xi$ with $\xi\in D$ satisfying $\xi < N$ it follows $f(x) < M.$

Definitions: 1 2 3 4

Explanations: 5

Lemmas: 6

Proofs: 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Propositions: 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

Theorems: 36

**Forster Otto**: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983**Kane, Jonathan**: "Writing Proofs in Analysis", Springer, 2016