Limit

Limit of Numerical Sequence

The most fundamental concept in calculus is that of a limit, it introduces the concept of approximation. The techniques used in calculus are based on approximation of a number, the limit $l$, arbritarily close. The limit $l$ of a sequence of numbers is the number to which the sequence comes as close a you like and remains at least that close.

We need to translate this concept of closeness into precise mathematical terms. Suppose $\delta$ is a small positive number. We say that the number $a_n$ is $\delta-$close to $l$, if the distance between $a_n$ and $l$ is less than $\delta$: \[ \norm{a_n-l}\less\delta \]

We need now also to translate in mathematical terms that the sequence of numbers need to stay that close. For that we need to have that for a certain item,let say the Nth item $a_n$ is $\delta-$close to $l$, and that this is true for all items following the Nth item: \[ \exists N \;\; \forall n \ge N \;\; \norm{a_n - l}\less \delta \]

If we add to this the notion "as close as you like" we have a general definition of a limiting process for a sequence of numbers: \[ \forall \delta >0 \exists N \;\; \forall n \ge N \;\; \norm{a_n - l}\less \delta \Leftrightarrow \lim_{n\to\infty} a_n = l \]

This mathematical definition of a limit or the convergence of a sequence requires that given any $\delta >0$ we must find an N which depends on $\delta$ and for each $n>N$ we must then show that $\norm{a_n - l}\less \delta$. We give an example of these type of proofs.

Example
Prove that \[ \lim_{n\to\infty} 1+\frac{1}{n}=1 \]
Solution

These type of proofs require that $l$ is already know beforehand and involve a symbolic approach and are difficult to computerize in this way. They were invented by mathematicians which had no computers available.

Limit of Function

Instead of a sequence of numbers we can also look at the evaluation of a function, $f(x)$ at a certain value $x=a$. Suppose we take $f(x)=x^2$ and $x=\pi$. If we approximate the square of $\pi$ with $x=\pi+\delta$ then we get an approximation which is off: $\epsilon=x^2-\pi^2=2\delta \pi +\delta^2$. We can make the difference $\epsilon$ arbritarily small, as long as we take $x$ close enough to $\pi$. This property can be stated formally as: \[ \forall \epsilon>0 \;\; \exists \delta>0 \;\; (\norm{x-\pi}\less \delta \implies \norm{x^2-\pi^2}\less \epsilon) \] This property implies $x\to \pi$ then $f(x)\to f(\pi)$ is called the limit of $f(x)$ at $x=\pi$ and denoted as: \[ \lim_{x\to \pi} f(x)=\pi^2 \]

Definition: Limit
Let $f:A\to\mathbb{R}^m,\;A\subset \mathbb{R}^n$ and $a$ a limit point of $A$ and $l \in \mathbb{R}^m$ we write \[ \lim_{x\to a} f(x)= l \] if and only if: \[ \forall \epsilon>0 \;\; \exists \delta>0 \;\; (\norm{x-a}\less \delta \implies \norm{f(x)-f(a)}\less \epsilon) \]

Note that this definition allows that $a \not \in A$, which means that the function is not defined at $a$. Also it is possible that $f(a)\neq l$. In both cases we say that the function is not continuous at $a$. A function is continuous, if and only if $\lim_{x\to a} f(x)=f(a)$.

We can recast the definition in terms of limits of sequences. Define the sequence $\{x_n\}$ such that $x_n\neq a$ and $\lim_{n\to \infty} x_n=a$ then we have: \[ \lim_{n\to \infty} f(x_n)=l \]

Differentiation

Differentiability relates to the approximation of a function, $\Delta f$, with a linear function, $\lambda(h)$ around a point $a+h$ of its domain.

Definition: Differentiation of a scalar function
A function $f:A\to\mathbb{R},\;A\subset \mathbb{R}$ is differentiable at $a\in\mathbb{R}$ if there exists a linear transformation $\lambda : \mathbb{R} \to \mathbb{R}$ defined by: \begin{equation} \lim_{h\to 0}\frac{f(a+h)-f(a)-\lambda(h)}{h}=0 \end{equation} with the linear transformation $\lambda(h)=m\cdot h$. The number $m$ is denoted by $\mathrm{D}f(a)$ or $f'(a)$ and called the derivative of $f$ at $a$ and the function $\lambda(h)+f(a)$ is the tangent line at $f$ through $a$.

Although higher dimensions functions are studied in the section on multivariate and vector calculus we develop the concept of differentiability here because the definition of differentiability has a simple generalization to higher dimensions.

Definition: Differentiation
A function $f:A\to\mathbb{R}^m,\;A\subset \mathbb{R}^n$ is differentiable at $a\in\mathbb{R}^n$ if there exists a linear transformation $\lambda : \mathbb{R}^n \to \mathbb{R}^m$ defined by: \begin{equation} \lim_{h\to 0}\frac{\norm{f(a+h)-f(a)-\lambda(h)}}{\norm{h}}=0 \end{equation} with linear transformation $\lambda(h)=M \cdot h$. The $m\times n$ matrix M is the Jacobian matrix of $f$ at $a$ and denoted by $Df(a)$ or $f'(a)$ and called the derivative of $f$ at $a$. The hyperplane $\lambda(a)+f(a)$ is tangent at $a$ to $f$.

Note \[ \lim_{h\to 0}\norm{f(a+h)-f(a)}=Df(a)\norm{h} \]

References



Copyright 2012 Jacq Krol. All rights reserved. Created September 2012; last updated September 2012.