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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 $\infty$.

Example :

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.

Let f(x) = $\left [ \frac{6x^{2}}{(3x^{2}+2x)} \right ]$, dividing both the numerator and the denominator by x

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 e

For the remaining four trigonometric functions the limit at infinity is undefined because all have a range of (

Let f(x) = x

Let f(x) = -2x

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.

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

For example, for f(x) = tan (x) the left hand side limit at odd multiples of $\frac{\pi }{2}$ is $\infty$

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 $\infty$

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 $\infty$.