The Definition of Limit

Let $f\left( x \right)$ be a real valued function if the value of the function $f\left( x \right)$ approaches a fixed number, say $L$, as $x$ approaches a number, say $a$. In this case we say that $L$ is the limit of function $f\left( x \right)$ as $x$ approaches $a$.

Mathematically, this can be written as:
$\mathop {\lim }\limits_{x \to a} f\left( x \right) = L$

We read it as “the limit of $f$ is $L$ as $x$ approaches $a$”.

If a variable $x$ assumes in succession a series of values
$1,\frac{1}{5},\frac{1}{{{5^2}}},\frac{1}{{{5^3}}},\frac{1}{{{5^4}}}, \ldots ,\frac{1}{{{5^n}}}$

Then $x$ is becoming smaller and smaller as $n$ increases and can be made as small as we please by making $n$ sufficiently large. This unending decrease of $x$ is mathematically written as $x \to 0$ and is read as $x$ tends to zero or $x$ approaches zero.