Key Rules and Examples of Limits to Infinity in Maths
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.
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.
From the above table, we can see that as x gets larger, 1/x tends to 0.
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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.
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.
FAQs on Limits to Infinity Explained for Students
1. What are limits at infinity in mathematics?
Limits at infinity describe the end behaviour of a function. It is a concept in calculus used to determine the value a function's output, f(x), approaches as its input, x, grows infinitely large (approaches +∞) or infinitely small (approaches -∞). This helps in understanding the long-term trend of the function's graph.
2. What does the notation lim x → ∞ f(x) = L signify?
This notation is a formal way of stating that as the variable x increases without bound (approaches infinity), the value of the function f(x) gets closer and closer to a finite number L. L is called the limit of the function at infinity. It essentially describes the value the function settles on as x becomes extremely large.
3. How do you find the limit at infinity for a rational function?
To find the limit of a rational function (a fraction of two polynomials) as x → ∞, you can compare the degrees of the numerator and the denominator. A common method is to divide every term in both the numerator and denominator by the highest power of x found in the denominator. The result depends on the degrees of the polynomials:
If the degree of the numerator is less than the degree of the denominator, the limit is 0.
If the degree of the numerator is equal to the degree of the denominator, the limit is the ratio of the leading coefficients.
If the degree of the numerator is greater than the degree of the denominator, the limit does not exist (it is ±∞).
4. What is the difference between a 'limit at infinity' and an 'infinite limit'?
This is a crucial distinction. A 'limit at infinity' describes what happens to f(x) when x → ∞ or x → -∞. The result is often a finite number. For example, lim x→∞ (1/x) = 0. In contrast, an 'infinite limit' describes a situation where the function's value f(x) grows without bound (approaches ∞ or -∞) as x approaches a specific finite number. For example, lim x→0 (1/x²) = ∞.
5. Why is infinity not treated as a real number when evaluating limits?
Infinity (∞) is not a number but a concept representing unbounded growth. It does not have a fixed value. Therefore, standard arithmetic operations like addition, subtraction, or division cannot be applied to it in the same way as real numbers. For instance, expressions like ∞ - ∞ or ∞/∞ are known as indeterminate forms because their value cannot be found without further analysis of the functions involved.
6. How are limits at infinity used to find the horizontal asymptotes of a function?
The concept of limits at infinity is directly used to define horizontal asymptotes. A horizontal line y = L is a horizontal asymptote of the graph of a function y = f(x) if either of the following is true:
lim x→∞ f(x) = L or lim x→-∞ f(x) = L.
Essentially, the horizontal asymptote is the value that the function approaches as its input x moves towards positive or negative infinity, representing the function's end behaviour.
7. When is it appropriate to use L'Hôpital's Rule for limits at infinity?
L'Hôpital's Rule is a powerful technique used to evaluate limits that result in an indeterminate form. For limits at infinity, it is applicable when taking the limit of a quotient of two functions, f(x)/g(x), as x → ∞, and the result is either ∞/∞ or 0/0. In such cases, the rule states that the limit of the original quotient is equal to the limit of the quotient of their derivatives, i.e., lim [f'(x)/g'(x)].
