$f$ will always reach 1 if $\lim \limits_{x \to \infty}f(x)=1^-$? Suppose an algorithm A, has success probability of 10%.
One experiment is composed by a (possibly infinite) number of executions of A.
In one experiment, the success probability of at least one execution after x trials can be found by: 
$P[x\ge1]=1-(\frac{90}{100})^x$ and so $\lim \limits_{x \to \infty}P[x\ge1]=1^-$
So, having that in mind, can we say that every experiment will always succeed?
I would say that we can't. If we say for example that the algorithm will succeed after k tries (for any value of k), then somebody could argue that after infinite number of experiments, then there would be at least one experiment in which the algorithm A failed after k tries. 
Or in another way, after infinite number of experiments, there will be at least one experiment in which A fails for infinite number of executions.
Is that right?

Let B be the algorithm running A until it succeeds.
There are different ways of looking at this


*

*There is a case where B runs forever (if A keeps failing)

*The probability of this case (that B runs forever), is $0$

*For any $k$, the probability, that B will require more than $k$ calls to $A$, is greater than $0$


So depending on what you want to know, the answer will be different.
