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.