# 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.