I'm working through a test question, in which I'm given the following statement:
If $\displaystyle \lim_{x\to \infty}f(x)=0$, then for large enough $x$,$f(x)$ gets closer and closer to $0$, without ever equalling $0$.
To that, I'm given some options and have to decide which is correct. The options of interest are as follows:
a) The statement is false, and $\displaystyle f(x) = \frac{\ln(x)}{x}$ is a counterexample.
b) The statement is false, and $\displaystyle f(x) = \frac{\sin x}{x}$ is a counterexample.
Clearly $\displaystyle \frac{\sin x}{x}=0$ every $\pi n$ radians ($n$ any integer), so this is an answer. Indeed, this is the correct answer to the question.
However, I propose $\displaystyle f(1)=\frac{\ln(1)}{1}=0$. Why isn't this a counterexample to the original statement?