Lesson Explainer: Limits at Infinity | Nagwa Lesson Explainer: Limits at Infinity | Nagwa

Lesson Explainer: Limits at Infinity Mathematics • Second Year of Secondary School

In this explainer, we will learn how to evaluate limits of a function when 𝑥 tends to infinity.

The limit of a function at infinity describes the behavior of the function’s output values as 𝑥 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 𝑥 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 𝑓(𝑡) to be the amount (in mass) of the radioactive element in the object at time 𝑡, this means that the value of 𝑓(𝑡) will approach zero as 𝑡 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 𝑓(𝑥) approach some finite value 𝐿 as the value of 𝑥 tends to infinity, then we say that the limit of 𝑓(𝑥) as 𝑥 approaches positive infinity exists and is equal to 𝐿 and we denote this as lim𝑓(𝑥)=𝐿.

If the values of 𝑓(𝑥) increase (or decrease) without bound as 𝑥 tends to infinity, then we say that the limit of 𝑓(𝑥) 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 𝑓(𝑥) and 𝑔(𝑥) be functions whose domains extend to positive infinity, and let 𝑐 be a nonzero constant. Then, the following identities hold as long as the right-hand side of the equation is not an indeterminate form, 00,,0, or : limlimlimlimlimlimlimlimlimlimiftheright-handsideiswelldenedlimlimlimiflim𝑐𝑓(𝑥)=𝑐𝑓(𝑥),(𝑓(𝑥)𝑔(𝑥))=𝑓(𝑥)𝑔(𝑥),(𝑓(𝑥)±𝑔(𝑥))=𝑓(𝑥)±𝑔(𝑥),(𝑓(𝑥))=𝑓(𝑥),𝑓(𝑥)𝑔(𝑥)=𝑓(𝑥)𝑔(𝑥)𝑔(𝑥)0.

Let us consider how the limit at infinity is portrayed in the graph of a function 𝑓(𝑥).

To find the limit of this function at infinity, we need to find the value 𝑓(𝑥) approaches as 𝑥 tends to infinity. This means that we are considering the 𝑦-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 𝑦-coordinates of the points on the graph lead to the number 2. This tells us that lim𝑓(𝑥)=2.

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 𝑓(𝑥) at infinity exists and is given by lim𝑓(𝑥)=𝐿.

Then, the graph 𝑦=𝑓(𝑥) has a horizontal asymptote 𝑦=𝐿.

Since we found earlier that lim𝑓(𝑥)=2, we can conclude that 𝑦=2 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 𝑥 grows larger. For instance, consider the table of values for the function 1𝑥 for large values.

𝑥1101001‎ ‎000
1𝑥10.10.010.001

From the table above, we can see that the value of the function 1𝑥 approaches zero as 𝑥 tends to positive infinity. This leads to lim1𝑥=0.

This limit also tells us that the graph 𝑦=1𝑥 has the horizontal asymptote 𝑦=0 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 𝑦=1𝑥.

From this graph, we can conclude that the 𝑦-coordinates of points on the graph approach 0 as 𝑥 tends to negative infinity. We can express this as lim1𝑥=0.

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 𝑥±. For instance, we have found that lim±1𝑥=0.

Also, for any constant 𝑎 and a positive integer 𝑛, we can apply the limit law for powers and scalar multiplications to write limlimlimlim𝑎𝑥=𝑎1𝑥=𝑎1𝑥=𝑎1𝑥=𝑎×0=0.

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 𝑎 and positive integer 𝑛, lim±𝑎𝑥=0.

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 𝑓(𝑥)=5𝑥+9𝑥2𝑥𝑥+11.

  1. Which of the following is equal to lim𝑓(𝑥)?
    1. lim11
    2. 𝑥lim
    3. 2𝑥lim
    4. 5𝑥lim
  2. Hence, find lim𝑓(𝑥).

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 𝑥 increases without bound. When we consider substituting a very large value for 𝑥, say 𝑥=1000000, the first term 5𝑥 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 𝑥, which is 𝑥. 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 5𝑥 at positive infinity.

We can make this idea more rigorous by justifying this behavior algebraically. Let us begin by factoring out 𝑥 from the polynomial. We can write 5𝑥+9𝑥2𝑥𝑥+11=𝑥5+9𝑥2𝑥1𝑥+11𝑥.

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 limlimlimlimlimlim5𝑥+9𝑥2𝑥𝑥+11=𝑥5+9𝑥2𝑥1𝑥+11𝑥.

We also recall that, for any real number 𝑎 and a positive integer 𝑛, lim𝑎𝑥=0.

This means that limlimlimlim9𝑥=0,2𝑥=0,1𝑥=0,11𝑥=0.

This leads to limlim5𝑥+9𝑥2𝑥𝑥+11=5𝑥.

This is option D.

Part 2

In the previous part, we found that the given limit at infinity is the same as 5𝑥.lim

This limit describes the behavior of the function 𝑥 as 𝑥 grows larger without bound. If we consider substituting larger values of 𝑥 into this expression, we can see that the resulting value will grow without bound. Recall that when the function value becomes larger as 𝑥 tends to infinity, we say that the limit of the function at infinity is equal to infinity. Hence, 5𝑥=5×=.lim

This means lim5𝑥+9𝑥2𝑥𝑥+11=.

In the previous example, we found the limit of a polynomial function at infinity by factoring out the highest power of 𝑥 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 𝑥. We can generalize this result for any polynomial by following a similar argument.

Rule: Limit at Infinity of a Polynomial Function

Let 𝑝(𝑥) be a polynomial function given by 𝑝(𝑥)=𝑎𝑥+𝑎𝑥++𝑎𝑥+𝑎,𝑎0.

Then limlim𝑝(𝑥)=𝑎𝑥.

This limit is equal to positive or negative infinity, if the sign of 𝑎 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 𝑥. 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 𝑓(𝑥)=3𝑥8𝑥92𝑥.

  1. Which of the following is equal to lim𝑓(𝑥)?
    1. 3892limlim
    2. 3+89+2limlim
    3. 3892lim
    4. 389+2lim
    5. 3892lim
  2. Find lim𝑓(𝑥).

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 3𝑥, while the leading term of the denominator is 2𝑥. Hence, the given rational function should behave the same at infinity as the quotient 3𝑥2𝑥, which can be reduced to a constant 32.

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 𝑥, which is 𝑥. This leads to 3𝑥8𝑥92𝑥=3𝑥8𝑥×(92𝑥)×=32.

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 limlimlim3𝑥8𝑥92𝑥=3892.

This is option A.

Part 2

In the previous part, we obtained that the given limit at infinity is equal to 3892.limlim

We recall that, for any real number 𝑎 and a positive integer 𝑛, lim±𝑎𝑥=0.

This means that limlim1𝑥=0,1𝑥=0.

Substituting these limits above, we obtain 38×09×02=32=32.

Hence, lim𝑓(𝑥)=32.

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 𝑥 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 𝑝(𝑥) and 𝑞(𝑥) be polynomials, and let 𝑚 be the degree of the denominator 𝑞(𝑥). To find the limit lim±𝑝(𝑥)𝑞(𝑥), we need to

  1. multiply the numerator and denominator of the quotient by 1𝑥,
  2. simplify the numerator and denominator of the quotient,
  3. apply the rule lim±𝑎𝑥=0 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 lim5𝑥92𝑥+5.

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 𝑥 in the denominator. In the given rational function, the highest power of 𝑥 in the denominator is 𝑥, so we can multiply the numerator and denominator of the quotient by 1𝑥. This leads to 5𝑥92𝑥+5=(5𝑥9)×(2𝑥+5)×=2+.

Applying the limit laws, we can write limlimlimlim5𝑥92𝑥+5=2+.

We recall that, for any real number 𝑎 and a positive integer 𝑛, lim𝑎𝑥=0.

This means that limlimlim5𝑥=0,9𝑥=0,5𝑥=0.

Substituting these limits above, we obtain lim5𝑥92𝑥+5=002+0=0.

Hence, lim5𝑥92𝑥+5=0.

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 lim𝑥7𝑥+3𝑥+7𝑥+48𝑥6𝑥6𝑥+4.

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 𝑥 for the denominator. In the given rational function, the highest power of 𝑥 in the denominator is 𝑥, so we can multiply the numerator and denominator of the quotient by 1𝑥. This leads to 𝑥7𝑥+3𝑥+7𝑥+48𝑥6𝑥6𝑥+4=𝑥7𝑥+3𝑥+7𝑥+4×(8𝑥6𝑥6𝑥+4)×=𝑥7+++8+.

Applying the limit laws, we can write limlimlimlimlimlimlimlim𝑥7𝑥+3𝑥+7𝑥+48𝑥6𝑥6𝑥+4=𝑥7+++8+.

We recall that, for any real number 𝑎 and a positive integer 𝑛, lim𝑎𝑥=0.

This means that limlimlimlimlim3𝑥=0,7𝑥=0,4𝑥=0,6𝑥=0,6𝑥=0.

Substituting these limits above, we obtain limlimlim𝑥7𝑥+3𝑥+7𝑥+48𝑥6𝑥6𝑥+4=𝑥78=𝑥+78.

We know that lim𝑥=; hence, lim𝑥+78=+78=.

This gives us lim𝑥7𝑥+3𝑥+7𝑥+48𝑥6𝑥6𝑥+4=.

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 𝑝(𝑥) and 𝑞(𝑥) be polynomials.

  • If 𝑝(𝑥) and 𝑞(𝑥) have the same degrees, then lim±𝑝(𝑥)𝑞(𝑥) is given by the ratio of the leading coefficients, which are the coefficients of the highest power of 𝑥 in both the numerator and denominator of the quotient.
  • If 𝑝(𝑥) has a lower degree than 𝑞(𝑥), then lim±𝑝(𝑥)𝑞(𝑥)=0.
  • If 𝑝(𝑥) has a higher degree than 𝑞(𝑥), then lim±𝑝(𝑥)𝑞(𝑥) 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 𝑎 and 𝑏, given that lim5𝑥2𝑥+3(𝑎+4)𝑥+(1𝑏)𝑥+5𝑥=.

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 𝑝(𝑥) and 𝑞(𝑥) be polynomials.

  • If degdeg𝑝(𝑥)=𝑞(𝑥), then lim±𝑝(𝑥)𝑞(𝑥) is equal to the ratio of the leading coefficients.
  • If degdeg𝑝(𝑥)<𝑞(𝑥), then lim±𝑝(𝑥)𝑞(𝑥)=0.
  • If degdeg𝑝(𝑥)>𝑞(𝑥), then lim±𝑝(𝑥)𝑞(𝑥)=±.

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 5𝑥2𝑥+3, 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 (𝑎+4)𝑥+(1𝑏)𝑥+5𝑥. If 𝑎+4 is nonzero, the degree of this polynomial would equal 6, which is greater than 5. Thus, we must have 𝑎+4=0, which leads to 𝑎=4. In this case, the first coefficient is equal to zero, which means that the denominator of our function is written as (1𝑏)𝑥+5𝑥. Similarly, if 1𝑏 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 1𝑏=0, which leads to 𝑏=1. This means that the denominator of our function is given by 5𝑥, whose degree is equal to 4. We note that since 5>4, 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, 𝑎=4,𝑏=1.

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 lim6𝑥+95𝑥+1.

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 𝑥 grows larger. Let us examine the behavior of the numerator and denominator of the quotient separately for larger values of 𝑥.

For the numerator, 6𝑥+9 is a square root of a polynomial function 6𝑥+9. We know that the polynomial grows like the leading term, which is the term with the highest power of 𝑥. In this case, the leading term of this polynomial is 6𝑥. Using this idea with the power rule for limits, we obtain limlimlimlimlim6𝑥+9=6𝑥+9=6𝑥=6𝑥=6𝑥.

This means that the numerator behaves like the function 6𝑥 as 𝑥 approaches infinity.

Next, consider the denominator 5𝑥+1. Since the denominator is a polynomial with the leading term 5𝑥, it behaves like 5𝑥 at infinity. This leads to the conclusion that the given quotient behaves like 6𝑥5𝑥, which simplifies to the constant 65.

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 𝑥. Hence, let us divide the numerator and denominator of this quotient by 𝑥. This leads to 6𝑥+95𝑥+1=6𝑥+9×(5𝑥+1)×=(6𝑥+9)×5+=6+5+.

Using limit laws, we can write limlimlim6𝑥+95𝑥+1=6+5+.

We recall that, for any real number 𝑎 and a positive integer 𝑛, lim𝑎𝑥=0.

This means that limlim1𝑥=0,1𝑥=0.

Substituting these limits above, we obtain lim6𝑥+95𝑥+1=65.

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 lim16𝑥5𝑥4𝑥, 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, 16𝑥5𝑥 and 4𝑥. Both these functions approach infinity as 𝑥 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 𝑎𝑏 is 𝑎+𝑏; hence, the conjugate of the given function, 16𝑥5𝑥4𝑥, can be written as 16𝑥5𝑥+4𝑥.

To simplify the given function, we begin by multiplying the function by a quotient whose numerator and denominator are equal to this conjugate expression: 16𝑥5𝑥4𝑥=16𝑥5𝑥4𝑥16𝑥5𝑥+4𝑥16𝑥5𝑥+4𝑥=16𝑥5𝑥4𝑥16𝑥5𝑥+4𝑥16𝑥5𝑥+4𝑥.

To multiply through the numerator of this fraction, we can use the difference of squares formula: (𝑎𝑏)(𝑎+𝑏)=𝑎𝑏. Since the square cancels out the square root of the first term, this expression simplifies to 16𝑥5𝑥(4𝑥)16𝑥5𝑥+4𝑥=16𝑥5𝑥16𝑥16𝑥5𝑥+4𝑥=5𝑥16𝑥5𝑥+4𝑥.

Now that we have simplified the given function, let us consider the limit at infinity. We can write limlim16𝑥5𝑥4𝑥=5𝑥16𝑥5𝑥+4𝑥.

We know that the limit at positive infinity describes the behavior of the function value when 𝑥 grows larger. Let us examine the behavior of the numerator and denominator of the quotient separately for larger values of 𝑥.

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 𝑥. In the denominator, we have a sum of a square root function, 16𝑥5𝑥, and a polynomial, 4𝑥. Although the square root expression contains the second-degree term 16𝑥, this term is under the square root, which means that it behaves like 16𝑥=4𝑥, which has the same degree as the polynomial term. Hence, the highest-degree term of the denominator is also 𝑥.

Thus, we can multiply the numerator and denominator of this quotient by 1𝑥, which leads to 5𝑥16𝑥5𝑥+4𝑥=5𝑥×16𝑥5𝑥+4𝑥×=5×(16𝑥5𝑥)+4=516+4.

Using limit laws, we can write limlim516+4=516+4.

We recall that, for any real number 𝑎, lim𝑎𝑥=0.

This means that lim5𝑥=0.

Substituting this limit above, we obtain limlim16𝑥5𝑥4𝑥=516+4=516+4=54+4=58.

Let us finish by recapping a few important concepts from this explainer.

Key Points

  • If the values of 𝑓(𝑥) approach some finite value 𝐿 as the values of 𝑥 tend to infinity, then we say the limit of 𝑓(𝑥) at infinity exists and is equal to 𝐿 and we denote this as lim𝑓(𝑥)=𝐿. The limit at negative infinity is defined similarly.
  • If the values of 𝑓(𝑥) increase (or decrease) without bound as 𝑥 tends to infinity, then we say that the limit of 𝑓(𝑥) 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: 00,,0, or .
  • A polynomial function at infinity behaves like its leading term, which is the term containing the highest power of 𝑥.
  • For any constant 𝑎 and positive number 𝑚, lim±𝑎𝑥=0.
  • Let 𝑝(𝑥) and 𝑞(𝑥) be polynomials, and let 𝑚 be the degree of the denominator 𝑞(𝑥). To find the limit lim±𝑝(𝑥)𝑞(𝑥), we need to
    • multiply the numerator and denominator of the quotient by 1𝑥,
    • simplify the numerator and denominator of the quotient,
    • apply the rule lim±𝑎𝑥=0 and find the answer.

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