calculating the function limit interatively - matlab

Question

How to calculate the limit of a function interatively in Matlab, by closer to the given limit value? The accuracy of closing is 10^(-7)
I suppose that that the taylor formula should be used, but don't know how to apply it there.
The function itself is :
The limit is 88.
In other words, the assignment is to present limits as series with assigned variables, compute them step-by-step, approach the limits' value with 10^(-7) precision.
example code of task:
syms x;
F=log(1+sin(x))/(sin(4*x));
a=limit(F,x,0);
disp(a)
sum=taylor(F,x,0,'Order',7);
disp(sum)
disp (subs(sum,x,0))

Answer 1

Calculating it with MATLAB is quite easy, when using the Symbolic Toolbox. The limit function is what you need:
syms x
limit((x^2-9*x-10)/(sqrt(x+6)-4),x,10)
ans =
88
If you want to calculate it by hand, you don't need Taylor series, you'll need L'Hopital's rule, which states
(image: wikipedia)
This leads to
To calculate this in MATLAB, you could use the diff function to get the derivative and do something like
syms x
f(x) = x^2-9*x-10;
g(x) = sqrt(x+6)-4;
r(x) = diff(f(x)) / diff(g(x));
r(10)
ans =
88
Well, as we are using MATLAB, we can of course just use Taylor series expansion and let MATLAB do the job. MATLAB has a taylor function which creates the Taylor expansion. As the Taylor expansion is exact around the expansion point and the error increases, the further away you are from that point, it is best to use 10 as expansion point.
syms x
t(x) = taylor((x^2-9*x-10)/(sqrt(x+6)-4),x,10,'Order',6);
t(10)
ans =
88

Answer 2

OK, now that I know what you're after, what you could perhaps do is use that taylor command and expand about a point that is quite far off from where you want to compute the limit. If we set the expansion point to be where you want to evaluate the limit, no matter what order polynomial you choose, you will get the correct result which is what I'm assuming you're not after.
Start at an expansion point that is far away, then keep incrementally increasing the order of polynomial of the Taylor series until you get your desired accuracy. You don't want to choose an expansion point that is too far away, or you will never get the right answer. As such, I'm going to expand at x = 7.
Something like this:
true_val = 88; %// Define true value
syms x;
f = (x^2-9*x-10)/(sqrt(x+6)-4); %// Define function
order = 2; %// Start with second order
format long g; %// For better formatting
while true %// Keep iterating...
% // Get Taylor polynomial centered at x = 7 of the current order
pol = taylor(f, x, 7, 'Order', order);
%// Evaluate the Taylor series
val = double(subs(pol, x, 10));
%// Show the results
disp(['Order: ' num2str(order)]);
disp('Result');
disp(val);
%// Check to see if we have at least 1e-7 accuracy then break out if yes
if abs(true_val - val) < 1e-7
break;
end
%// Increment the order by 1
order = order + 1;
end
This is what I get:
Order: 2
Result
86.9892652074553
Order: 3
Result
88.0453290425764
Order: 4
Result
87.9954798755339
Order: 5
Result
88.0005926106152
Order: 6
Result
87.9999105029301
Order: 7
Result
88.0000147335223
Order: 8
Result
87.999997429935
Order: 9
Result
88.0000004672668
Order: 10
Result
87.9999999123696