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

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