Prove that the following integral is divergent $$\int_0^\infty \frac{7x^7}{1+x^7}$$
Im really not sure how to even start this. Does anyone care to explain how this can be done?

An inproper integral will diverge if the limit of the function at infinity is not zero (as Chris pointed out, it's a different business if the limit doesn't exist). Here, 
$$
\lim_{x\to\infty}\frac{7x^7}{1+x^7}=7,
$$
so the integral diverges.
