Limits at Positive Infinity

So far we have discussed limits at some fixed numbers. In this section we shall be concerned with limits at positive infinity.

Let us consider the function f defined by the equation f\left( x \right) = \frac{{2{x^2}}}{{{x^2} + 1}}. Let x take on the values 0, 1, 2, 3, 4, 5, 10, 100, 1000 etc., allowing x to increase without bound. The corresponding function values are given in the table below.

x
0
1
2
3
4
5
10
100
1000
f\left( x \right) = \frac{{2{x^2}}}{{{x^2} + 1}}
0
1
\frac{8}{5}
\frac{9}{5}
\frac{{32}}{{17}}
\frac{{25}}{{13}}
\frac{{200}}{{101}}
\frac{{20000}}{{10001}}
\frac{{2000000}}{{1000001}}

When the independent variable x is increasing without bound through positive values, as in the above table, we say that x is approaching positive infinity and write it as x \to + \infty .

We see from the given table that x \to + \infty the function values f\left( x \right) get closer and closer to 2, f\left( x \right) \to 2. Because of this, we say

\mathop {\lim }\limits_{x \to + \infty } \frac{{2{x^2}}}{{{x^2} + 1}} = 2