MATLAB: Error when trying to calculate a limit with double variable.

Question

When I execute the following code :

n=linspace(10,10e20);

un=(1+1./n).^n;

limit(un,n,inf);

It returns the following error: Undefined function 'limit' for input arguments of type 'double'.

can anyone help me get the value of the limit on matlab ?

Answer 1

limit( ) operates on symbolic expressions. E.g.,

>> syms n
>> limit((1+1/n)^n,inf)
ans =
exp(1)

For the numerical approximation, you can't use a number that is too large or the sum will have too much cancellation error. Double precision has about 15 digits, so you should stay under that to get any meaningful results. E.g.,

>> n = 1e10;
>> (1+1/n)^n   % <-- 1/n contributes to the sum 1+1/n
ans =
   2.718282053234788   % <-- reasonable approximation
>> exp(1)
ans =
   2.718281828459046
>> n = 1e20;
>> (1+1/n)^n   % <-- 1/n does not contribute to the sum 1+1/n
ans =
     1   % <-- bad approximation