cardinality of the set of real functions that approaches $0$ when $x\longrightarrow \infty$ I know that this set is a subset of the set of all real functions. Hence its cardinality is less than or equal to $\aleph ^\aleph=2^\aleph$.
The question is how do I prove the second direction?
(actually I'm not sure that $2^\aleph $ is the cardinality of the above set)
 A: First of all you are missing a subscript on that $\aleph$. Second of all the set of all functions from the reals to $2$ is going to be $2^{\mathfrak{c}}=2^{(2^{\aleph_0})}$. Where $\aleph_0=|\mathbb{N}|$.
Now in answer to your question the number of functions that converge to $0$ at infinity will be the maximum possible, that is the same as the number of functions from the reals to the reals without any restriction. You just need to think about how many functions from $[0,1]$ to $[0,1]$ there are. You can obviously extend any such function to have a limit of $0$ at infinity.
Edit
Just to explain a bit about what the $\aleph$-notation means so you can see why you need some subscripts. $\aleph$'s are infinite cardinalities in increasing size. The first such cardinality is denoted $\aleph_0$ and is the size of the least infinite cardinal which is actually generally denoted $\omega$. The second least infinite cardinal is $\omega_1$ which has cardinality $\aleph_1$. While very often in set theory people will not bother distinguishing $\aleph$'s and the corresponding $\omega$'s it's probably better to think of them as separate and whenever you're actually talking about the cardinality of a set you should the $\aleph$-notation (as you did) whereas whenever you are talking about the cardinal you should use the $\omega$-notation
One last note: I should probably point out what that strange $\mathfrak{c}$ is just in case. $\mathfrak{c}$ is the standard symbol used for the cardinality of the continuum or in other words the cardinality of the real numbers/powerset of the natural numbers. This might or might not be equal to $\aleph_1$, but some parts of set theory are somewhat boring if it is. Still in general the most "probable" choices for $\mathfrak{c}$ seem to be $\aleph_1$ (under the Continuum Hypothesis) or $\aleph_2$ (under the Proper Forcing Axiom) or possibly some unnamed largish cardinal if you want to play around with cardinal invariants of the continuum.
