Concept of Limit

Meaning of the Phrase “Tend to Zero”:
Suppose a variable ‘x’ assumes in succession a set of values.

1,\;\frac{1}{{10}},\;\frac{1}{{{{10}^2}}},\;\frac{1}{{{{10}^3}}},\;\frac{1}{{{{10}^4}}},\; \cdots ,\;\frac{1}{{{{10}^{\text{n}}}}},\; \cdots

Clearly, ‘x’ is becoming smaller and smaller as n is increasing and can be made as small as we want by making ‘n’ sufficiently large. This un-ending decrease of ‘x’ is symbolically expressed by “{\text{x}} \to 0” and read as “x tends to zero”.
It should be noted that the symbol “{\text{x}} \to 0” is quite different from {\text{x}} = 0. The equation ${\text{x}} = 0$ means that ‘x’ has actually assumed the value zero while {\text{x}} \to 0 implies that the variable ‘x’ takes in succession a series of values which become smaller and smaller such that the difference between x and 0 become and remain less than any pre-assigned positive number, however small.

Meaning of the Phrase “x Tends to Infinity”:
Let a variable ‘x’ assume the values

1,\;10,\;{10^2},\;{10^3},\;{10^4},\; \cdots ,\;{10^{\text{n}}},\; \cdots

It is evident from the above values that ‘x’ will get larger and larger values as ‘n’ increases and can be made ‘n’ sufficiently large.
This unending increase of ‘x’ is symbolically expressed by “{\text{x}} \to \infty ” and is read as “x tends to infinity”. This simply indicates an endless progress of x to numerical greatness without band.

Meaning of the Phrase “x Tends to a”:
Suppose ‘x’ assumes in succession a set of values.

{\text{a}} + \frac{1}{{10}},\;{\text{a}} + \frac{1}{{{{10}^2}}},\;{\text{a}} + \frac{1}{{{{10}^3}}},\; \cdots ,\;{\text{a}} + \frac{1}{{{{10}^{\text{n}}}}},\; \cdots

Clearly, the successive difference of these values from ‘a’ is

\frac{1}{{10}},\;\frac{1}{{{{10}^2}}},\;\frac{1}{{{{10}^3}}},\;\frac{1}{{{{10}^4}}},\; \cdots ,\;\frac{1}{{{{10}^{\text{n}}}}},\; \cdots


which become smaller and smaller as ‘n’ increases and can be made small as we like by making ‘n’ sufficiently large.
This behavior of ‘x’ is symbolically expressed by “{\text{x}} \to {\text{a}}” which implies that the variable x takes in succession a set of values that approaches nearer and nearer to ‘a’ in such a manner that the numerical value of {\text{x}} \to {\text{a}} remains less than any pre-assigned positive number, however small.