Limits at Infinity

Limits at infinity is an important topic in calculus. It deals with the maximization of any application that has to be dealt with in physics, chemistry and engineering applications.

Example :
A particle inside a potential well varies from x $\rightarrow$ 0 to $\infty$.

Finding Limits at Infinity

Finding limits at infinity is an important concept as it helps us understand the behavior of a function. A function can have a domain from +$\infty$ to -$\infty$ or can have a range from +$\infty$ to -$\infty$. Limits at infinity has a broad meaning. It may refer to the limit of the function when the variable approaches $\pm \infty$ , the function approaches $\pm \infty$ for some values of the variable or when the variable approaches $\pm \infty$ , the function also approaches $\pm \infty$ . In the first case it would be referred to as the limits at infinity, in the second case it is called as the infinite limits and in the third case it is referred to as infinite limits at infinity.

Limits at Infinity Rules

To make the evaluation of the limits at infinity, a closer study has revealed certain interesting facts and helped mathematicians to frame the limits at infinity rules.
Before we proceed, we would like to remind you that limits at infinity are relevant only in case of rational functions. For polynomial functions, the quick answer is the limit would either be $\infty$ or -$\infty$.

Let f(x) is a rational function expressed in the form f(x) = $\frac{g(x)}{h(x)}$ $h(x)\neq 0$. The order of the functions play an important role.

The limit at infinity rules state that,

1) If the order of g(x) is higher than that of h(x) then the limit at infinity is $\infty$ , if the sign of the ratio of the leading coefficients is positive and -$\infty$ if the sign of the ratio of the leading coefficients is negative.

2) If the order of g(x) is same as that of h(x) then the limit at infinity is  the ratio of the leading coefficients.

3) If the order of g(x) is lower than that of h(x) then the limit at infinity is  0.

Evaluating Limits at Infinity

Evaluating limits at infinity for rational functions is easily done by the rules we have already stated. These rules are framed by just simple algebraic operations.  As an example, let us illustrate one case.

Let f(x) = $\left [ \frac{6x^{2}}{(3x^{2}+2x)} \right ]$, dividing both the numerator and the denominator by x2, f(x) = $\frac{6}{[3+(\frac{2}{x})]}$. As x ->$\infty$, the term $\frac{2}{x}$ becomes 0 and hence the limit is $\frac{6}{3}$ = 2, which is  the ratio of the leading coefficients.

In case of polynomial functions the limit at infinity is $\infty$ or -$\infty$, depends on the sign of the leading coefficient. In case of logarithmic functions, the limit at infinity is $\infty$ .

In case of exponential functions, the limit at infinity is $\infty$or 0, depending on the sign of the exponent variable.
For example, the limit at infinity of ex is $\infty$ and of e-x is 0.The limit at infinity of trigonometric functions are interesting. Since the range of sine and cosine functions are restricted to [-1,1], the limit is indefinite but it is bounded only between -1 and 1.

For the remaining four trigonometric functions the limit at infinity is undefined because all have a range of (- $\infty$,$\infty$).

Limits at Infinity Examples

 Let us see few examples of limits at infinity,

Let f(x) = x4 - 5x2 + 6. This is a polynomial function and the leading coefficient is positive. Hence the limit at infinity of this function is $\infty$ .

Let f(x) = -2x3 - 4x2 + 8. This is a polynomial function and the leading coefficient is negative. Hence the limit at infinity of this function is -$\infty$.

Let f(x) = $\left [ \frac{3x}{(4x^{2}+5x)} \right ]$. This is a rational function and the order of the numerator function is less than that of the denominator function.
 
As per the limits at infinity rules, the limit at infinity for this function is 0.

We have already explained about the limits at infinity for logarithmic, exponential and trigonometric functions. We will see another important example, probably, the definition of e, the exponential constant.

Consider the function f(x)= $[1+(\frac{1}{x})]^{x}$. Let us evaluate the limit at infinity of this function.

Taking the natural logarithm on both sides,

In f(x) = In F(x) = $ [1+(\frac{1}{x})]^{x}$  = $x*f(x)$ = $[1+(\frac{1}{x})]$

= $x*[(\frac{1}{x})-(\frac{1}{2x^{2}})+(\frac{1}{3x^{3}})-.....]$

                      =$ [(1)-(\frac{1}{2x})+(\frac{1}{3x^{2}})] $

or, f(x) = $e^{[(1)-(\frac{1}{2x})+(\frac{1}{3x^{2}})-.....]}$

Now the limit at infinity  of f(x) = limit at infinity  of $e^{[(1)-(\frac{1}{2x})+(\frac{1}{3x^{2}})-.....]}$ = e

Therefore, limit at infinity  of $[1+(\frac{1}{x})]^{x}$  = e.

Infinite Limits

Infinite limits means the limit of the function is $\infty$ or -$\infty$ for one or more values of the variable. Obvious examples are logarithmic functions and trigonometric functions, other than sine and cosine functions.

For rational functions, infinite limits occur at points where the denominator function becomes 0.

A logarithmic function has infinite limit of  - $\infty$ for logarithm of a term approaches 0. Trigonometric functions, other than sine and cosine functions have infinite limits of $\infty$ or -$\infty$ for some values of the variable.
For example, for f(x) = tan (x) the left hand side limit at odd multiples of $\frac{\pi }{2}$ is $\infty$ and the right hand side limit at odd multiples of $\frac{\pi }{2}$ is -$\infty$.

Finding Infinite Limits

There are different methods of finding infinite limits of functions, depending on the types of functions.

The infinite limits of rational functions occur at the zeroes of the denominator function. Hence the method to find infinite limits is to find the zeroes of the function. It may be noted that if there are common factor terms between the numerator and the denominator functions, the limit is not infinite for the solution of the common factor. The rational function has only a hole there.
For example if $f(x)$ = $\frac{(x^{2}-1)}{[(x-1)(x+2)]}$, the limit at x = 1 is not infinite. On the other hand, it has an infinite limit at x = -2.The infinite limits of other functions can be found by solving for the value of the variable which would make the function approach $\infty$ or - $\infty$. We will discuss more with examples in the next section.

Evaluating Infinite Limits

Evaluating infinite limits or solving infinite limits involve more of algebraic techniques based on the concept of infinite limits. Let us illustrate with a few examples.   

Infinite Limits at Infinity

Functions which tend to $\infty$ or -$\infty$ when the variable approaches $\infty$ or -$\infty$, they are said to have infinite limits at infinity. All polynomial functions, logarithmic function and exponential functions with positive exponents have infinite limits at infinity.

A rational function in which the numerator function is of higher order than that of the denominator function has infinite limits at infinity. Because the rational function can be simplified by long division and the quotient will contain the variable or its higher powers,
For example, the limit at infinity for the function f(x) = $\frac{(x^{2}+2x-4)}{(x)}$ is $\infty$.