#f(x) = e^x#

#e^x# is a transcendental function meaning that is both irrational and cannot be expressed in terms of a finite sequence of the algebraic operations of addition, multiplication, and root extraction.

Hence, #e^x# can never (except in the trivial case #x=0#) be expressed as a fraction, the root of any polynomial with rational coeffients or the sum of any finite series. Thus, it can only ever be approximated by a number of any base.

Several definitions of #e^x# exist. Two of the most well known of these are:

#e^x = lim_(n->oo) (1+x/n)^n# (limit known to exist #forall x in RR#)

#e^x = sum_(n=0) ^oo (x^n)/(n!)# (sum known to converge #forall x in RR#)

From the second definition above we can approximate #e^3.2# as a decimal as follows:

#e^3.2 = 1 + 3.2 + 3.2^2/(2!)+ 3.2^3/(3!) + 3.2^4/(4!) + .......#

# approx 24.53253#