Limits at infinity and infinite limits

What is the limit?

The limit of a function f(x) can be defined as the value; the function f(x) approaches as the value of x approaches a certain x = k value. We represent the limit of the function f(x) as the value of x tends to k as,

xkf(x)= C

Here, the value of C is called the limit of the function as x tends to k.

We use limits in the cases where the value of a function f(x) cannot be defined at a particular value of x. For example, let’s take a look at the function, f(x) = (x2 – 3x + 2)/(x-2).

The value of this function is not defined at x=2. So now we take the limit,

x2(x2 – 3x + 2)/(x-2)

We can factorize the function as x2(x-2)(x-1)/(x-2)

This gives us the limit of f(x) as x tends to 2 to be 1.

Here one may ask the question of why we did not factorize the function and divide it with denominator. But here we have to consider the fact that, when we substitute the value x = 2 for f(x), we get f(2) = (2-2)(2-1)/(2-2)

⇒ f(2) = (0/0) x 1.

The factor (0/0) is not defined. But a value small enough that it is very close to zero will give the division (x/x) the value 1. And that is the concept we use here.

There are instances where we will have to deal with cases involving infinity. We might get the value of the limit to be infinity, or a constant value for a limit that tends to infinity.

Let us take a deeper look into these instances.

Limit at infinity

When we talk about the value of a function f(x) for different values of x, we can see that the value of x can give the function different values at different rates as we increase x. But there are instances where the value of f(x) might not change much, no matter how significant the change in the value of x is. This means that the value of the function might tend to a constant value as the value of x tends to reach +∞ or -∞.

We can represent this in the following way,

 x∞f(x)= C 

Here the value of f(x) tends to be constant value, C, as the value of x approaches infinity.

We have the same situation for the cases where the value of the function tends to a constant value as x tends to -∞.

x-∞g(x)= M

Here the value of g(x) tends to a constant value M as the value of x tends to -∞

In order to understand this concept, let’s take a look at function f(x) given by,

f(x) = (4×2 + 1)/(7×2 + 6). We need to find the value of this function as x tends to infinity.

If we substitute ∞ for x directly, we might get (∞/∞) which is not defined. So we take the limit of the function as x tends to infinity. This gives,

x∞f(x) = x∞(4x2 + 1)/(7x2 + 6)  

⇒ x∞(4 + 1/x2)/(7 + 6/x2)   ( ∵ dividing both sides by x2)

⇒ (x∞4 + x∞(1/x2))/(x∞7 + x∞(6/x2)) = 4/7

Here the limit for constant values is the constant itself, and x∞(1/x2) is zero.

Definition of limit at infinity

For a function f(x), we can say that the limit x∞f(x)=M exists only when for all k >0 , there exists an S > 0 such that for all x > S we have | f(x) – M | < k. 

Note: We say that the limit of function f(x) exists at infinity when the function tends to a particular value and x increases unbounded.

Infinite limits

For certain functions, let’s say a function f(x), the value of the function might blow up to infinity as the value of x reaches a particular value. This situation gives rise to what we call infinite limits.

We can represent it in the following way,

 xkf(x)= ∞

Here the value of the function reaches infinity as the value of x tends to a certain value, k

We can also have cases where the value of a function can tend to -∞ for a certain value of x like,

xkG(x)= -∞

For example, let’s take a look at the function, f(x) = 1/x2; the value of the function blows up to infinity as the value of x goes closer to zero.

So we write  x0(1/x2) = ∞

Note: The value of a function can tend to infinity as x approaches a specific value, and it may become a real number for values less than or more significant than that particular value of x. For example, 1/(x – 2) tends to infinity for x = 2, but it has real values for x = 1 and x = 3.

 Conclusion.

The limit of a function f(x) at infinity is the value that the function tends to as the value of x increases unbounded. This limit exists only when the converged value of the function is a real number. Infinite limit is when the value of the function g(x) tends to infinity as x reaches a certain value. These limits help understand the nature of a function in extreme situations, like when the value of x becomes huge or tiny.