Infinite Limits Function

I. Mathematical Expression:

$$ f(x)=\frac{\sin(x)}{x}$$

where:

\(\bullet\) \(-100 \leq x \leq 100\)

\( \rhd\) For Minimization Mode:

\(\bullet\) It has two global minima: \(f_{min}(x^*)=-0.217233628211222\)

\(\bullet\) \(x^*=\pm 4.493409471849579\) (determined by us using MapleSoft 2015)

\( \rhd\) For Maximization Mode:

\(\bullet\) It has one global maxima: \(f_{max}(x^*) \rightarrow\) approaches 1

\(\bullet\) \(x^* \rightarrow\) approaches zero from \(-ve\) and \(+ve\) sides

\(\bullet\) Please note that at \(x=0 \rightarrow f(0)=\frac{0}{0}\) which means indeterminate.

II. Citation Policy:

If you publish material based on databases obtained from this repository, then, in your acknowledgments, please note the assistance you received by using this repository. This will help others to obtain the same data sets and replicate your experiments. We suggest the following pseudo-APA reference format for referring to this repository:

Ali R. Al-Roomi (2015). Unconstrained Single-Objective Benchmark Functions Repository [https://www.al-roomi.org/benchmarks/unconstrained]. Halifax, Nova Scotia, Canada: Dalhousie University, Electrical and Computer Engineering.

Here is a BiBTeX citation as well:

@MISC{Al-Roomi2015,
author = {Ali R. Al-Roomi},
title = {{Unconstrained Single-Objective Benchmark Functions Repository}},
year = {2015},
address = {Halifax, Nova Scotia, Canada},
institution = {Dalhousie University, Electrical and Computer Engineering},
url = {https://www.al-roomi.org/benchmarks/unconstrained}
}

III. 2D-Plot:

IV. MATLAB M-File:

% Infinite Limits Function
% Range of initial points: -100 <= x <= 100
% Two global minima: x=-4.493409471849579,4.493409471849579; and 
% One global maxima: as x approaches zero from -ve and +ve sides
% Be note that when x=0, f(x)=0/0 >> indeterminate
% fmin(x)=-0.217233628211222 and fmax(x) approaches 1
% Coded by: Ali R. Alroomi | Last Update: 17 May 2015 | www.al-roomi.org
 
clear
clc
warning off
 
xmin=-100;
xmax=100;
R=100000; % steps resolution
x=xmin:(xmax-xmin)/R:xmax;
 
for i=1:length(x)
    f(i)=sin(x(i))/x(i);
end
 
plot(x,f,'r','LineWidth',2);grid;set(gca,'FontSize',12);axis([-100 100 -0.5 1.25]);
xlabel('x','FontName','Times','FontSize',20,'FontAngle','italic');
ylabel('f(x)','FontName','Times','FontSize',20,'FontAngle','italic');

Click here to download m-file

V. References:

[1] SageMath, "Calculus Tutorial," [Accessed July 08, 2015]. [Online]. Available: http://www.sagemath.org/calctut/inflimits.html
[2] Ali R. Alroomi, "The Farm of Unconstrained Benchmark Functions," University of Bahrain, Electrical and Electronics Department, Bahrain, Oct. 2013. [Online]. Available: http://www.al-roomi.org/cv/publications