# 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.

 $x$ $0$ $- 1$ $- 2$ $- 3$ .adslot_1 { width: 336px; height: 280px; } @media (max-width:350px) { .adslot_1 { width: 300px; height: 250px; } } $- 4$ $- 5$ $- 10$ $- 100$ $- 1000$ $f\left( x \right) = \frac{{2{x^2}}}{{{x^2} + 1}}$ $0$ $1$ .adslot_1 { width: 336px; height: 280px; } @media (max-width:350px) { .adslot_1 { width: 300px; height: 250px; } } $\frac{8}{5}$ $\frac{9}{5}$ $\frac{{32}}{{17}}$ $\frac{{25}}{{13}}$ $\frac{{200}}{{101}}$ $\frac{{20000}}{{10001}}$ $\frac{{2000000}}{{1000001}}$

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.