infinity times infinitesimal - what happens? So what happens if we multiply infinite number by. Infinitesimal number? Like $dx \times \infty$ where $dx$ is treated as in one-dimensional integration.

Also, can we divide infinite number by infinite number and get a finite number?
In the ordinary calculus, there are no infinitesimals. 

Abraham Robinson and others, from the $1950$'s on, developed non-standard analysis, which does have infinitesimals, and also "infinite" number-like objects, that one can work with in ways that are closely analogous to the way we deal with ordinary real numbers. 

In non-standard analysis, an infinitesimal times an infinite number can have various values, depending on their relative sizes. The product can be an ordinary real number. But it can also be infinitesimal, or infinite. Similarly, the ratio of two "infinite" objects in a non-standard model of analysis can be an ordinary real number, but need not be.

The calculus can be developed rigorously using Robinson's infinitesimals. There are even some courses in calculus that are based on non-standard models of analysis. Some have argued that this captures the intuition of the founders of calculus better than the traditional limit-based approach.

For further reading, you may want to start with the Wikipedia article on Non-standard Analysis.
This is a question for surreal numbers. Surreal numbers are a really amazing thing invented by John Conway that include numbers like 0 and 3/4, but also things like "twice the square root of infinity, all plus an infinitesimal". This question depends on the values of the infinite and infinitesimal, but the way it works is this. The number ω is defined as the number of items in the set {0,1,2,3,4,5...}, so it's infinite. The number ε is defined as 1/ω. So ω*ε is obviously one. If you think about it a bit, it makes sense that 2ε^2 * ω is 2ε, and so on.

 http://en.wikipedia.org/wiki/Surreal_number