Meaning of 'Tends' $\rightarrow$ Meaning of 'Tends' $\rightarrow$.
I found a definition of tends in the book Higher Algebra by Barnard and Child and I am being obtuse. 

"Definition: To say that x tends to zero is to say that x varies in such a way that its numerical value becomes and remains less than any positive number that we may choose, no matter how small." 

Comment on the definition stated and explain the fact. 
 A: One need not dwell on the "definition" of the separate meaning of a symbol in, especially, mathematics. 
The symbol $dx$ appearing in an integral does not have meaning in itself, unless one talks about differential forms.
In the same token,
the symbol "$\to$" is context-determined, that is to say, writing "$\to$" alone does not say any concrete thing. To exemplify this, just recall that we use "$\to$" both in the context of specifying functions and that of limits. 
The best way, to me, to "define" the phrase "$x \to a$" in a context involving limits is to say that it is simply a suggestive, mnemonic abbreviation for rigorous analytic language. 
For example, we say "$f(x) \to l$ as $x \to a$" to mean "for every $\varepsilon > 0, \cdots$ $|f(x) - l| < \varepsilon \cdots $    
A: Before getting into the formal definition of tends to and a limit, we can try to see what happens when we bring x arbitrarily close to a, i.e. as $x \to a$.
Consider a simple function $f(x) = (x^2 - 1)/(x - 1)$.
If we try to work out the value of $f(x)$ at $x = 1$, we get $0/0$ which is indeterminate.
What we can do though, is try to 'approach' 1.
$f(0.5) = 1.5$
Well, we are quite far, let's move closer.
$f(0.9) = 1.9$
Good, but not close enough for me.
$f(0.99) = 1.99$
Well, that looks better.
$f(0.999) = 1.999$
Now we're getting somewhere. 
This can go on and on and I can get the value of $f(x)$ arbitrarily close 2.
$f(0.9999) = 1.9999$
$f(0.99999) = 1.99999$
We cannot say what $f(x)$ is when $x = 1$. But we see that $f(x)$ is getting closer and closer to $2$ as $x \to 1$. This is formally written as
$\lim_{x \to 1} f(x) = 2$
The limit of $f(x)$ as $x \to 1$ is $2$.
