Let's define a few symbols and a function (which is just an expression depending on x):

In [1]: var('x z')
Out[1]: (x, z) 
In [2]: f = 1/(1+x**2)

Let's evaluate this function at 1:

In [3]: f.subs(x, 1)
Out[3]: 1/2

We can compute the derivative of this function:

In [4]: diff(f, x)
Out[4]: -2*x/(x**2 + 1)**2

What is f's limit to infinity? (Note the double o (oo) for the infinity symbol):

In [5]: limit(f, x, oo)
Out[5]: 0

Here's how to compute a Taylor series (here, around 0, of order 9). The Big O can be removed with the removeO() method.

In [6]: series(f, x0=0, n=9)
Out[6]: 1 - x**2 + x**4 - x**6 + x**8 + O...