Limits at Negative Infinity

So far we have been discussed limits at some fixed numbers; in this section we shall be concerned with the limits at negative 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 decreasing through negative values without bound. In this case, we say that x is approaching negative infinity and write x \to - \infty as shown in the table gives the corresponding function values, f\left( x \right). The corresponding function values are given in the given 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}}

We note that the function values are the same for the negative numbers as for the corresponding positive numbers. So, we can see that f\left( x \right) \to 2 as x \to - \infty . i.e.

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