1. F distribution

How can you show that the f distribution goes to zero as x approaches infinity

How can you show that the f distribution goes to zero as x approaches infinity
If it didn't then the area under the curve could never equal 1 (or any other finite number for that matter).

Where are you having trouble in using the pdf to establish this result?

3. show that the limit is zero

show that the limit is zero
Yes, I realise what you want to do. What I'm asking is where are you stuck in using the pdf to do this?

5. Sketch the graph of the F density function given in Exercise 1. In particular, show that
1. f(x) at first increases and then decreases, reaching a maximum at the mode
x
= (m - 2)/(m(n + 2)).
2. f(t) converges to 0 as t approaches infinity.

The pdf has the form $f(x) = C \, \frac{x^{\frac{n}{2} - 1}}{(M + nx)^{\frac{n+m}{2}}}$ where $C$ is a constant.
1. Solve $f'(x) = 0$. I suggest using the quotient rule to get the derivative.
2. I suggest you first substitute some concrete values for $m$ and $n$ and then attempt taking the limit. This will let you get the feel of things.