Limits

Overview

Let $p$ be a point. An open interval containing $p$ as its midpoint is called a neighborhood of $p$. That is, a neighborhood of $p$ consists of all real $x$ satisfying $p-r<x<p+r$ for some $r>0$. The number $r$ is called the radius of the neighborhood. A neighborhood of $p$ is usually denoted as $N(p)$ or, if specifying radius $r$, as $N(p;r)$.

Let $A$ be a real number and $f$ a function defined on some neighborhood of a point $p$ (except possibly at $p$). Then the limit of $f(x)$, as $x$ approaches $p$, is equal to $A$ is denoted as

and means that for every neighborhood $N_1(A)$, there is a neighborhood $N_2(p)$ such that

Epsilon-Delta

The $ϵ,\delta$-terminology is an alternative formulation of the definition of limits, in terms of the radii of the neighborhoods. Let $A$ be a real number and $f$ a function defined on some neighborhood of a point $p$ (except possibly at $p$). Then

means that for every $ϵ>0$, there exists a $\delta >0$ such that

One-Sided Limits

If $f$ has a limit $A$ at $p$, then it also has a left- and right-hand limit at $p$, both of these being equal to $A$. If though the left- and right-hand limit do not equal one another at $p$, we say the limit of $f$ at $p$ does not exist.

Left-Hand Limit

Let $A$ be a real number and $f$ a function defined on some neighborhood of a point $p$ (except possibly at $p$). The left-hand limit of $f(x)$, as $x$ approaches $p$, is equal to $A$ is denoted as

and means that for every neighborhood $N_1(A)$, there is a neighborhood $N_2(p)$ such that

Right-Hand Limit

Let $A$ be a real number and $f$ a function defined on some neighborhood of a point $p$ (except possibly at $p$). Then the right-hand limit of $f(x)$, as $x$ approaches $p$, is equal to $A$ is denoted as

and means that for every neighborhood $N_1(A)$, there is a neighborhood $N_2(p)$ such that

Infinity

By Domain

The symbolism

means that for every number $ϵ>0$, there exists another number $M>0$ such that

An analagous definition exists as $x\to -\mathrm{\infty }$.

By Range

The symbolism

means that for every number $M>0$, there exists a number $\delta >0$ such that if $0<|x-a|<\delta$ then $f(x)>M$. The left- and right-handed limits are defined similarly.

Analagous definitions exist for $f(x)\to -\mathrm{\infty }$ as $x\to a$.

Algebraic Limit Theorem

The following identities assume $f$ and $g$ are functions such that

• $\underset{x\to p}{lim}[f(x)+g(x)]=A+B$
• $\underset{x\to p}{lim}[f(x)-g(x)]=A-B$
• $\underset{x\to p}{lim}[f(x)\cdot g(x)]=A\cdot B$
• If $B\ne 0$, $\underset{x\to p}{lim}[f(x)÷g(x)]=A÷B$

Squeeze Theorem

Suppose that $f(x)\le g(x)\le h(x)$ for all $x\ne p$ in some neighborhood $N(p)$. Suppose also that

Then $g(x)\to A$ as $x\to p$.

Asymptotics

o-notation

Assume $g(x)\ne 0$ for all $x\ne a$ in some interval containing $a$. Notation $f(x)=o(g(x))$ as $x\to a$ means that

It follows that:

• $o(g(x))\pm o(g(x))=o(g(x))$ as $x\to a$.
• $o(cg(x))=o(g(x))$ as $x\to a$.
• $f(x)\cdot o(g(x))=o(f(x)g(x))$ as $x\to a$.
• $o(o(g(x)))=o(g(x))$ as $x\to a$.

Indeterminate Forms

A limit of a composed function is said to take on an indeterminate form if the limit cannot be computed using the algebraic limit theorem. There are seven different forms:

1. $0/0$
2. $\mathrm{\infty }/\mathrm{\infty }$
3. $0\times \mathrm{\infty }$
4. $\mathrm{\infty }-\mathrm{\infty }$
5. $0^0$
6. $1^{\mathrm{\infty }}$
7. ${\mathrm{\infty }}^0$

L'Hôpital's Rule

Assume $f$ and $g$ have derivatives $f^{\prime }(x)$ and $g^{\prime }(x)$ at each point $x$ of an open interval $(a,b)$ and suppose that

or

Assume also that $g^{\prime }(x)\ne 0$ for each $x\in (a,b)$. If the limit of $f^{\prime }(x)/g^{\prime }(x)$ as $x\to a^{+}$ exists, then the limit of $f(x)/g(x)$ as $x\to a^{+}$ also exists and

An analagous theorem also holds for left-handed limits. Combining the left- and right-handed variants yields the two-sided result of the same kind in which $x\to a$ in an unrestricted fashion.

Exponential Bounds

If $a>0$ and $b>0$, then

In other words, $(\ln x{)}^a=o(x^b)$ and $x^a=o(e^{bx})$.