In this explainer, we will learn how to evaluate limits of a function when tends to infinity.
The limit of a function at infinity describes the behavior of the function’s output values as tends to infinity. Unlike the limit of a function at a finite point, the direct substitution method is not a valid method for these limits since infinity is not a number. Instead, we need to consider the behavior of the function value as becomes larger without bound.
Let us consider a real-life example regarding a radioactive element in an object. We know that radioactive elements decay (exponentially) over time; hence, the radioactive element in an object gradually disappears over time. If we denote to be the amount (in mass) of the radioactive element in the object at time , this means that the value of will approach zero as tends to positive infinity. This motivates the definition of limit at infinity.
Let us formally define limit at infinity.
Definition: Limit at Infinity
If the values of approach some finite value as the value of tends to infinity, then we say that the limit of as approaches positive infinity exists and is equal to and we denote this as
If the values of increase (or decrease) without bound as tends to infinity, then we say that the limit of at infinity is equal to positive (or negative) infinity respectively.
We note that all the limit laws regarding the limit of the sum, difference, product, and quotients of a pair of functions apply exactly the same way for the limit at infinity.
Rule: Limits Laws
Let and be functions whose domains extend to positive infinity, and let be a nonzero constant. Then, the following identities hold as long as the right-hand side of the equation is not an indeterminate form, , or :
Let us consider how the limit at infinity is portrayed in the graph of a function .
To find the limit of this function at infinity, we need to find the value approaches as tends to infinity. This means that we are considering the -coordinates of the points on the graph as we move toward the right edge of the graph. Following the given graph in this manner, the -coordinates of the points on the graph lead to the number 2. This tells us that
This limit at infinity is closely related to a horizontal asymptote of the graph of the function, which is a horizontal line the graph approaches as we move to the right or the left side of the graph.
Definition: Limit at Infinity and Horizontal Asymptotes
Say that the limit of function at infinity exists and is given by
Then, the graph has a horizontal asymptote
Since we found earlier that , we can conclude that is a horizontal asymptote of this function as shown below.
We can also understand the limit at infinity by looking at the table of function values as grows larger. For instance, consider the table of values for the function for large values.
| 1 | 10 | 100 | 1 000 | |
| 1 | 0.1 | 0.01 | 0.001 |
From the table above, we can see that the value of the function approaches zero as tends to positive infinity. This leads to
This limit also tells us that the graph has the horizontal asymptote as we can see below.
Although we will mostly consider limits at positive infinity, we should keep in mind that the limits at negative infinity can be similarly defined and computed. For limits at negative infinity, we need to follow the points on the graph as we move toward the left edge of the graph.
For instance, we can consider the left part of the graph .
From this graph, we can conclude that the -coordinates of points on the graph approach 0 as tends to negative infinity. We can express this as
We note that the limit at negative infinity of this function is the same as the limit at positive infinity. When the limits at both positive and negative infinities are the same, we can write the limit as . For instance, we have found that
Also, for any constant and a positive integer , we can apply the limit law for powers and scalar multiplications to write
We can see that the conclusion will be the same when we apply the limit at negative infinity. This leads to a more general rule, which we will use to solve various limit-at-infinity problems.
Rule: Limit at Infinity of Reciprocal Functions
For any real number and positive integer ,
This rule is very useful in finding the limit at infinity for a wide variety of functions, as we will see. To portray the main idea for applying this rule, we will consider the limit at infinity of a polynomial function in our first example.
Example 1: Evaluating Limits of Polynomials at Infinity
Consider the polynomial .
- Which of the following is equal to ?
- Hence, find .
Answer
Part 1
In this part, we need to find the limit of a polynomial at infinity. We recall that the limit of a function at infinity describes the behavior of the function values as increases without bound. When we consider substituting a very large value for , say , the first term will have the largest magnitude out of the five terms in this polynomial. This is because this term contains the factor with the highest power of , which is . In comparison to this term, the other four terms in this polynomial will be negligible in size. This leads to the idea that the limit of this function at infinity will behave like the limit of at positive infinity.
We can make this idea more rigorous by justifying this behavior algebraically. Let us begin by factoring out from the polynomial. We can write
We recall that the limit laws apply to the limit at infinity the same way. Using the limit laws regarding the sum, difference, and product of a pair of functions, we can write
We also recall that, for any real number and a positive integer ,
This means that
This leads to
This is option D.
Part 2
In the previous part, we found that the given limit at infinity is the same as
This limit describes the behavior of the function as grows larger without bound. If we consider substituting larger values of into this expression, we can see that the resulting value will grow without bound. Recall that when the function value becomes larger as tends to infinity, we say that the limit of the function at infinity is equal to infinity. Hence,
This means
In the previous example, we found the limit of a polynomial function at infinity by factoring out the highest power of from the polynomial and applying the limits of reciprocal functions at infinity. As a result, we found that the limit of this polynomial is the same as the limit of the leading term, which is the term containing the highest power of . We can generalize this result for any polynomial by following a similar argument.
Rule: Limit at Infinity of a Polynomial Function
Let be a polynomial function given by
Then
This limit is equal to positive or negative infinity, if the sign of is positive or negative respectively.
An important idea coming from this rule is that a polynomial at infinity grows like the leading term, or the term containing the highest power of . Using this idea, we can also find the limit of a rational function at infinity, as we will see in the next example.
Example 2: Evaluating Limits of Rational Functions at Infinity
Consider the rational function .
- Which of the following is equal to ?
- Find .
Answer
Part 1
In this part, we need to find the limit of a rational function at negative infinity. Since a rational function is a quotient of polynomials, we can find this limit by considering the property of polynomials on the numerator and denominator of the rational function. We recall that the limit at infinity of a polynomial is controlled by the leading term, or the term with the highest power. In the numerator of the given rational function, the leading term is , while the leading term of the denominator is . Hence, the given rational function should behave the same at infinity as the quotient , which can be reduced to a constant .
Let us make this idea more rigorous by using algebra. We begin by dividing the numerator and denominator of the quotient by the highest power of , which is . This leads to
We recall that the limit laws apply to the limit at infinity the same way. Using the limit laws regarding the difference and quotient of a pair of functions, we can write
This is option A.
Part 2
In the previous part, we obtained that the given limit at infinity is equal to
We recall that, for any real number and a positive integer ,
This means that
Substituting these limits above, we obtain
Hence,
In the previous example, we found the limit of a rational function by dividing the numerator and denominator of the quotient by the highest power of and applying the limit at infinity of reciprocal functions. This method can be applied for many different functions when finding the limit at infinity.
How To: Finding the Limit at Infinity of a Rational Function
Let and be polynomials, and let be the degree of the denominator . To find the limit , we need to
- multiply the numerator and denominator of the quotient by ,
- simplify the numerator and denominator of the quotient,
- apply the rule and find the answer.
We note that we multiply the numerator and denominator of the quotient by the reciprocal of the highest power of the denominator. In the previous example, this did not affect our method since both the numerator and denominator of the quotient had the same degree. When we have a rational function with different degrees, it is better to multiply by the reciprocal of the highest power of the denominator, to avoid the situation when we end up with a zero in the denominator.
In the next example, we will consider the limit at infinity of a rational function where the numerator and denominator of the quotients are polynomials of different degrees.
Example 3: Finding the Limit of a Rational Function at Infinity
Find .
Answer
In this example, we need to find the limit at infinity of a rational function. We recall that to find the limit of a rational function, we can begin by multiplying the numerator and denominator of the quotient by the reciprocal of the highest power of in the denominator. In the given rational function, the highest power of in the denominator is , so we can multiply the numerator and denominator of the quotient by . This leads to
Applying the limit laws, we can write
We recall that, for any real number and a positive integer ,
This means that
Substituting these limits above, we obtain
Hence,
In the next example, we will find the limit at infinity of a rational function where the numerator has the higher degree.
Example 4: Finding the Limit of a Rational Function at Infinity
Find .
Answer
In this example, we need to find the limit at infinity of a rational function. We recall that to find the limit of a rational function at infinity, we can begin by multiplying the numerator and denominator of the quotient by the reciprocal of the highest power of for the denominator. In the given rational function, the highest power of in the denominator is , so we can multiply the numerator and denominator of the quotient by . This leads to
Applying the limit laws, we can write
We recall that, for any real number and a positive integer ,
This means that
Substituting these limits above, we obtain
We know that ; hence,
This gives us
In previous examples, we found the limit at infinity of different rational functions. Examining our method closer leads to the following general conclusion.
Rule: Limits at Infinity of Rational Functions
Let and be polynomials.
- If and have the same degrees, then is given by the ratio of the leading coefficients, which are the coefficients of the highest power of in both the numerator and denominator of the quotient.
- If has a lower degree than , then .
- If has a higher degree than , then is equal to positive or negative infinity.
We could have used this property to solve the previous examples faster. While this is a useful rule to keep in mind, finding the limit algebraically is applicable in a wider variety of problems. Getting used to the algebraic method of finding the limit at infinity will also lead to a more concrete understanding of this subject.
In the next example, we will apply this rule to identify unknown constants in a function from the given limit at infinity.
Example 5: Finding Unknowns in a Rational Function given Its Limit at Infinity
Find the values of and , given that .
Answer
In this example, we are given the limit of a rational function at infinity. We know that the limit at infinity of a rational function depends on the degrees of the polynomials in the numerator and denominator of the function. We recall the rule for the limit at infinity of rational functions.
Let and be polynomials.
- If , then is equal to the ratio of the leading coefficients.
- If , then .
- If , then .
In particular, we note that the limit at infinity is equal to a finite number for the first two cases. Since the limit at infinity in this example is infinite, our rational function must belong to the third case. That is, the degree of the polynomial in the numerator is greater than the degree of the polynomial in the denominator. Let us consider the degree of these polynomials in our function.
The numerator of our function is , which is a polynomial of degree 5. Hence, the degree of the polynomial in the denominator must be less than 5. The denominator of our function is . If is nonzero, the degree of this polynomial would equal 6, which is greater than 5. Thus, we must have which leads to . In this case, the first coefficient is equal to zero, which means that the denominator of our function is written as . Similarly, if is nonzero, the degree of this polynomial is equal to 5, which is the same degree as the numerator. This cannot be true based on the given limit at infinity. Hence, we must have which leads to . This means that the denominator of our function is given by , whose degree is equal to 4. We note that since , the degree of the numerator is greater than the degree of the denominator. This places our function in the third category in the stated rule above, whose conclusion agrees with the given limit at infinity.
Hence,
We have considered the limits at infinity of polynomials and rational functions. In these examples, we used the fact that a polynomial at infinity behaves like its highest-degree term. This idea was helpful for evaluating the limits at infinity of rational functions.
Similar strategy can be used in a function where either the numerator or the denominator contains a radical root. In this case, rather than selecting the highest-degree term (which may lie under a root), we need to identify the overall behavior of the numerator and denominator considering the root. We will consider this in the next example.
Example 6: Finding the Limit of a Combination of Root and Polynomial Functions at Infinity
Find .
Answer
In this example, we need to find the limit at infinity of a quotient. We know that the limit at positive infinity describes the behavior of the function value when grows larger. Let us examine the behavior of the numerator and denominator of the quotient separately for larger values of .
For the numerator, is a square root of a polynomial function . We know that the polynomial grows like the leading term, which is the term with the highest power of . In this case, the leading term of this polynomial is . Using this idea with the power rule for limits, we obtain
This means that the numerator behaves like the function as approaches infinity.
Next, consider the denominator . Since the denominator is a polynomial with the leading term , it behaves like at infinity. This leads to the conclusion that the given quotient behaves like , which simplifies to the constant .
We can make this argument more rigorous by using algebra. We saw that both the numerator and denominator of the quotient behave like a constant times . Hence, let us divide the numerator and denominator of this quotient by . This leads to
Using limit laws, we can write
We recall that, for any real number and a positive integer ,
This means that
Substituting these limits above, we obtain
In the previous example, we found the limit at infinity of a function whose numerator contained the square root function. We can use the same strategy to find the limit at infinity of a function that contains the difference of square roots. At first sight, these problems many not look similar since it is not given in form of a quotient. But by multiplying by the conjugate of the square root expression, we can write these functions as a quotient, which makes the previously established method available. We will consider such a limit in our final example.
Example 7: Finding the Limit of Root Functions at Infinity Using Rationalization
Determine , if it exists.
Answer
In this example, we need to find the limit at infinity of a function. Our function is given as a difference of two functions, and . Both these functions approach infinity as approaches infinity, which means that this limit can be symbolically written . This is a type of an indeterminate form, which means that we are unable to determine the value of this limit based on the current form. To find the limit of a function in an indeterminate form, we must algebraically simplify the given function until we are able to evaluate the limit.
Since the given function is the difference of a square root function and a polynomial, we can think of the conjugate method, which is often used to simplify such algebraic expressions. Recall that the conjugate of expression is ; hence, the conjugate of the given function, , can be written as
To simplify the given function, we begin by multiplying the function by a quotient whose numerator and denominator are equal to this conjugate expression:
To multiply through the numerator of this fraction, we can use the difference of squares formula: . Since the square cancels out the square root of the first term, this expression simplifies to
Now that we have simplified the given function, let us consider the limit at infinity. We can write
We know that the limit at positive infinity describes the behavior of the function value when grows larger. Let us examine the behavior of the numerator and denominator of the quotient separately for larger values of .
To find the limit at infinity of a quotient function, we multiply the numerator and denominator by the reciprocal of the highest-degree term. The numerator of the quotient is a polynomial, where the highest-degree term is . In the denominator, we have a sum of a square root function, , and a polynomial, . Although the square root expression contains the second-degree term , this term is under the square root, which means that it behaves like , which has the same degree as the polynomial term. Hence, the highest-degree term of the denominator is also .
Thus, we can multiply the numerator and denominator of this quotient by , which leads to
Using limit laws, we can write
We recall that, for any real number ,
This means that
Substituting this limit above, we obtain
Let us finish by recapping a few important concepts from this explainer.
Key Points
- If the values of approach some finite value as the values of tend to infinity, then we say the limit of at infinity exists and is equal to and we denote this as The limit at negative infinity is defined similarly.
- If the values of increase (or decrease) without bound as tends to infinity, then we say that the limit of at infinity is equal to positive (or negative) infinity.
- Limit laws apply the same ways to the limit at infinity, as long as the right-hand side of an identity does not result in an indeterminate form: , or .
- A polynomial function at infinity behaves like its leading term, which is the term containing the highest power of .
- For any constant and positive number ,
- Let and be polynomials, and let
be the degree of the denominator . To find the limit , we need to
- multiply the numerator and denominator of the quotient by ,
- simplify the numerator and denominator of the quotient,
- apply the rule and find the answer.
