Limits of Rational Functions

What is a rational function?

A rational function is defined as the ratio of two polynomial functions, where the polynomial in the denominator is not zero. 

Say P(x) and Q(x) are polynomial functions, and f(x)is a rational function in the form of P(x)Q(x), given that Q(x) is a non-zero polynomial. Then

f(x) =P(x)Q(x), where Q(x)is not equal to zero. 

With Q(x)= 1, every polynomial function becomes rational. Those functions that cannot be expressed in the form of a polynomial, say, f(x) =Tan(x), are not rational functions. 

Do you know that the concept of a rational function is applied in several parts of our day-to-day activities?

Application of rational functions

• Calculating speed using the relationship between speed, distance, and time
• Calculating the density of an object using mass and volume
• Calculating the amount of work a person can do in the given amount of time. 

What is a limit in Math?

A limit denotes the value that a function approaches as the input reaches some value. The concept of limit is an integral part of calculus as it is primarily used to define continuity, derivatives, and integrals. 

Consider a function. f(x) = x2

When you put this in a graph, you can see that when the value of the function, f(x) = x2, approaches zero, the value of x also approaches zero. 

This is defined as if xl, f(x)a then we say that a is the limit of function f(x) and is commonly denoted as

limxl f(x) = 1.

Because there exist two ways in which l can approach a value, i.e. from the right or the left, all values of x around l could take a value less than l or greater than l. 

When x moves from right, we call it the right-hand limit, denoted by

xl+f(x)

On the other hand, when  x moves from left, it is called the left hand-limit denoted by 

 xl-f(x)= 1.

Applying limits to rational functions

xaf(x) =xa P(x)Q(x)

= xaP(x)xaQ(x)

=P(a)Q(a)

Calculating the limits of rational functions helps us predict how the graph of that function would behave at asymptotes. Finding the limits of rational functions can be simple at times and may require tricky methods sometimes. Following are some methods used to find the limits of rational functions – 

• Substitution method

In this method, we determine limits by substituting the value that x is approaching in the function. 

For example, 

x1x3+4x +20

To find the limit of the rational function, f(x), as x approaches 1, we can directly substitute the value of 1 for x in the equation. 

= x1   x3+4 a1 x+20

= 13+41+20

= 521

• Factoring method

Consider the following function, 

f(x) = x2-4x+2.

How will you calculate the limit of rational function f as x approaches two? 

When you try to solve this using the substitution method explained above,

x-2×2-4x+2. = x-2   x2-4 x-2 x+2

= (-2)2-4-2+2

=00

Here, the numerator and denominator become zero, and paradoxes follow. 

So, to evaluate problems of this kind, we follow a factoring method. 

In the factoring method, we factor the numerator and denominator. Here, in the above problem, the denominator is factored already. 

When we write the numerator in its factorial form, we arrive at the equation below, 

x2-4 = x2-22

    = (x+2)(x-2)

x-2×2^-4x+2  = x-2(x+2)(x-2)x+2

x+2 cancels, and we now have, 

x-2(x-2)= (-2-2)

    = -4

The value of the limit of the function is -4 when x tends to approach -2.

• Conjugate method

Consider the below problem. 

x5x+20-5x-5

Try to find the limit of a rational function using the substitution method. 

x5x+20-5x-5 = 5+20-5 5-5

    = 25-5 5-5

    = 25-5 5-5

    = 00

The result indicates that problems of the above kind that involve solving the limits of rational functions require an altogether different approach and method.

For such cases, we begin by multiplying both the numerator and denominator by the conjugate of the given radical expression, x+20-5, which is  x+20+5

Now the above-given function would become, 

x5x+20-5x-5*(x+20+5)(x+20+5)

x5(x+20)2-(5)2(x-5)(x+20+5)

x5(x+20)-(25)(x-5)(x+20+5)

x5(x-5)(x-5)(x+20+5)

Cancel (x-5) in the numerator and denominator, and the function would become, 

x51(x+20+5)

= 1(5+20+5)

= 1(25+5)

= 1(5+5)

= 110

Conclusion

Thus, we conclude the formal definition of limits as

‘a’ is the limit of function f(x) as x approaches l if, for every number >0, there will be a corresponding >0 for all values of x.

 xlf(x) = a

If and only if, 0<|x-l| <  |f(x)-a| < 

From the above, we can conclude that the right-handed limit of the function is, 

0

xl+f(x) = a

And the left-handed limit is defined as,

–

xl-f(x) = a

On the whole, we evaluate the functions in the numerator and denominator of the rational function at the said points. If this is of type 00, we try to cancel the factors that reduce the value of the limit to be of form 00. We do this by rewriting the function using the factorial method and conjugate method.