Symmetric limits I read that you can use something called a "symmetric limit" to evaluate the improper integral of sin(x) as 0, something that I attempted by taking the right hand improper integral and adding it to the L.H. improper integral... Only the limits don't exist. Could someone explain how a "symmetric limit" can be put to practical use?

For improper Riemann integral over $(-\infty,\infty)$. The symmetric limit means
taking the limit of corresponding proper Riemann integral symmetrically.
$$\int_{-\infty}^{\infty} f(x) dx\;\stackrel{def}{=}\;\lim_{R\to\infty} \int_{-R}^R f(x)dx$$
You can view this as a special case of taking the Cauchy principal value of corresponding integral.
Note please note that above is not the general definition of improper Riemann integral. However, I'm not aware of any standard notation to distinguish the symmetric limit version from the general version of definition.
