Limits To Infinity

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Limits To Infinity in Detail

Sometimes we can't find the actual value of a function directly, but we can see what it should be as we get closer and closer to that particular value.  A limit is a value that a function (or a sequence) "approaches" as the input (or index) "approaches" some value.


Limits

Let’s understand this with an example.

Consider the function x2 - 1/ x - 1

We know that the given function is not defined when the value of x is 1, because division by zero is not a valid mathematical operation. So, we can find value as x approaches 1.

Let’s try approaching when x tends to 1.


x

x2 - 1/ x - 1

0.5

1.50

0.9

1.90

0.99

1.99

0.999

1.999

0.9999

1.9999


From the above example, we can see that when x gets close to 1, then the value of x2 - 1/ x - 1 gets close to 2.

When x = 1, we don’t know the value (as it is indeterminate form). We can see that value gets close to 2. We can give answers as 2 but it is not the actual value. Hence the concept of the word “limit” came into existence.

The value of x2 - 1/ x - 1 as x approaches 1 is 2.


Infinity:

Something that is boundless or endless or else something that is larger than any real number is known as infinity. It is denoted by a symbol ∞.

Let’s understand with an example.

Find the value of one divided by infinity (1/∞).

Infinity is not a defined value. So, 1/∞ is similar to 1/smart.

We could say that 1/∞ = 0, but how it is possible because if we try to divide 1 into infinite pieces they can end up to 0 each. You may think about what happened to 1.

In fact, the value of 1/∞ is known to be undefined.


So, instead of trying to find it for infinity. Let’s try it for a larger value.


x

1x

1

1.00000

2

0.50000

4

0.25000

10

0.10000

100

0.01000

1000

0.00100

10000

0.00010


From the above table, we can see that as x gets larger, 1/x tends to 0.


(Image will be uploaded soon)


We can conclude two fact:

  • We can’t say what happens to x when it tends to infinity.

  • We can see that 1/x is getting close to 0.

Hence the limit of 1/x as x approaches infinity is 0.  We can write it as

lim (1/x) = 0 when x approaching ∞.

In a mathematical way, we are not talking about when x = ∞, but we know the value as x gets bigger the value gets closer and closer to 0.

So, infinity can’t be used directly but we can use the limit.


Limits to Infinity:

How to find the limit of a function as x approaches infinity?

Let function be y = 2x.


x

2x

1

2

2

4

4

8

10

20

100

200

...

...


So, from the above table, we can say that as x approaches infinity, then 2x also approaches infinity.

Don’t consider “=” sign as the exact value in the limit. We can’t actually get to infinity, but in limit language the limit is infinity.


Infinity and Degree

  • Functions like 1/x approaches to infinity. This is also valid for 1/x2 and so on.

  • A function such as x will approach infinity, same we can apply for 2x or  x/9, and so on. Likewise functions with x2 or x3 etc will also approach infinity.

  • We should be careful with negative functions like -x will approach -infinity. So we have to look at the sign of x and then decide the function value.

Let’s understand negative function value with an example:

Consider 2x2 - 4x

Sol: We know that 2x2 will tend towards +infinity and -5x will tend towards -infinity. But x2 value will be larger as compared to x. So 2x2 - 4x will tend to +infinity.

When we look for the degree of the function, check the highest exponent in the function.


The degree of function is divided into two parts:

  • The degree is greater than 0, the limit is infinity.

  • The degree is less than 0, the limit is 0.

Rational Function

A rational function is one that is the ratio of two polynomial functions.

Let f(x) = P(x)/Q(x)

P(x) = x3 + 2x - 1 and Q(x) = 3x2

Now compare the degree of P(x) to the degree of Q(x).

  • If the degree of P is less than the degree of Q the limit value is 0.

  • If the degree of P and Q are the same divide the coefficient of terms with the largest exponent.

  • If the degree of P is greater than the degree of Q then two cases come.

  • The limit value is positive infinity.

  • Or maybe negative infinity. We need to look at the sign and then decide.

FAQ (Frequently Asked Questions)

1. What are Infinity Limits?

Ans: Limits at infinity can be described as behaviour of the function as the independent variable increases or decreases without the bound. As we know that is not the actual value, we can only find function approaches that value.

2. Define the Additive Property of Infinity.

Ans: As per additive property, if anything or any number is added to infinity then the sum is also equal to infinity. As infinity is a huge number, adding or subtracting some value will not change any value resultant will also be infinity.  Ex- 20 + ∞ = ∞.