1.15 Connecting Limits at Infinity and Horizontal Asymptotes

5 min read•january 31, 2023

Anusha Tekumulla

ethan_bilderbeek

Anusha Tekumulla

ethan_bilderbeek

AP Calculus AB/BC ♾️

279 resources

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Limits at Infinity

In this section, we will focus on understanding the behavior of functions as they approach infinity in the x-direction. We will explore the concept of limits at infinity and how they relate to horizontal asymptotes in a function.

Objectives:

Interpret the behavior of functions using limits involving infinity in the x-direction.
Understand the concept of limits at infinity.
Describe and explain asymptotic and unbounded behavior of functions using limits.

Essential Knowledge:

The concept of a limit can be extended to include limits at infinity in the x-direction.
Asymptotic and unbounded behavior of functions in the x-direction can be described and explained using limits at infinity.

When a function approaches a value of positive or negative infinity in the x-direction, the limit is said to be infinite. For example, if a function f(x) approaches positive infinity as x approaches positive infinity, we would write the limit as:

lim x->+infinity f(x) = +infinity

Similarly, if a function approaches negative infinity as x approaches negative infinity, we would write the limit as:

lim x->-infinity f(x) = -infinity

In both cases, the limit does not exist in the traditional sense, as the function does not approach a specific value. Instead, the limit tells us about the behavior of the function as x approaches infinity.

Horizontal asymptotes are another way to describe the behavior of a function as it approaches infinity in the x-direction. A horizontal asymptote occurs when the function approaches a constant value as x approaches positive or negative infinity.

For example, a function with a horizontal asymptote at y = a would be written as:

f(x) = a + 1/x

As x approaches positive or negative infinity, the denominator (1/x) becomes smaller and smaller, causing the value of the function to approach the constant value of a. This results in a horizontal asymptote at y = a.

Looking Further

To further expand on the concept of limits at infinity and horizontal asymptotes, it is important to understand the role of leading terms in a function. The leading term of a function is the term with the highest degree of x. This term determines the behavior of the function as x approaches positive or negative infinity.

For example, in the function f(x) = x^2 + x - 6, the leading term is x^2. As x approaches positive or negative infinity, x^2 becomes much larger than the other terms, causing the function to approach positive infinity. This is why the limit of the function as x approaches positive or negative infinity is +infinity.

In contrast, in the function g(x) = 1/x, the leading term is 1/x. As x approaches positive infinity, 1/x becomes smaller and smaller, causing the function to approach 0. This is why the limit of the function as x approaches positive infinity is 0 and the function has a horizontal asymptote at y = 0.

Finally, it is important to understand that while a function may have a limit at infinity or a horizontal asymptote, this does not guarantee that the function will approach this value. For example, the function h(x) = x^2 + sin(x) approaches positive infinity as x approaches positive infinity, but the function will oscillate between positive and negative values, so it does not have a horizontal asymptote.

Limits at infinity and horizontal asymptotes provide a powerful tool for analyzing the behavior of functions in the x-direction as x approaches positive or negative infinity. By understanding the role of leading terms, we can describe and explain asymptotic and unbounded behavior in a concise and effective way.

Examples

Let's look at some examples to further illustrate the concept of limits at infinity and horizontal asymptotes:

f(x) = x, as x approaches positive infinity, the function approaches positive infinity, and we would write the limit as: lim x->+infinity f(x) = +infinity This function has no horizontal asymptote
g(x) = x^2, as x approaches positive infinity, the function approaches positive infinity, and we would write the limit as: lim x->+infinity g(x) = +infinity This function has no horizontal asymptote
h(x) = x^2-2x-3, as x approaches positive and negative infinity, the function approaches positive and negative infinity respectively, and we would write the limit as: lim x->+infinity h(x) = +infinity, lim x->-infinity h(x) = -infinity This function has no horizontal asymptote
j(x) = x^3+3x^2+3x+1/(x+1), as x approaches positive infinity, the function approaches 1, and we would write the limit as: lim x->+infinity j(x) = 1 As x moves closerto positive infinity, the denominator (x+1) becomes larger and larger, causing the value of the function to approach the constant value of 1. This results in a horizontal asymptote at y = 1.
k(x) = 1/x, as x approaches positive infinity, the function approaches 0, and we would write the limit as: lim x->+infinity k(x) = 0 As x moves closer to positive infinity, the denominator becomes larger and larger, causing the value of the function to approach 0. This results in a horizontal asymptote at y = 0.
l(x) = 1/x^2, as x approaches positive infinity, the function approaches 0, and we would write the limit as: lim x->+infinity l(x) = 0 As x moves closer to positive infinity, the denominator becomes larger and larger, causing the value of the function to approach 0. This results in a horizontal asymptote at y = 0.
m(x) = x^2+x-6/x+2, as x approaches positive infinity and -2, the function approaches 1 and -3 respectively, and we would write the limit as: lim x->+infinity m(x) = 1, lim x->-2 m(x) = -3 As x moves closer to positive infinity and -2, the denominator becomes larger and larger, causing the value of the function to approach 1 and -3 respectively. This results in horizontal asymptotes at y = 1 and y = -3.
n(x) = x^2-5x+6/x^2-4x+4, as x approaches positive and negative infinity, the function approaches 1 and -1 respectively, and we would write the limit as: lim x->+infinity n(x) = 1, lim x->-infinity n(x) = -1 As x moves closer to positive and negative infinity, the denominator becomes larger and larger, causing the value of the function to approach 1 and -1 respectively. This results in horizontal asymptotes at y = 1 and y = -1.
p(x) = x^3-9x+3/x^2+1, as x approaches positive and negative infinity, the function approaches 0, and we would write the limit as: lim x->+infinity p(x) = 0, lim x->-infinity p(x) = 0 As x moves closer to positive and negative infinity, the denominator becomes larger and larger, causing the value of the function to approach 0. This results in horizontal asymptote at y = 0.
q(x) = (x^2-4)/(x^2-16), as x approaches positive and negative infinity, the function approaches 0, and we would write the limit as: lim x->+infinity q(x) = 0, lim x->-infinity q(x) = 0 As x moves closer to positive and negative infinity, the denominator becomes larger and larger, causing the value of the function to approach 0. This results in a horizontal asymptote at y = 0.

Summary

In summary, understanding limits at infinity and horizontal asymptotes is crucial for interpreting the behavior of functions as they approach infinity in the x-direction. By using limits and analyzing the behavior of functions, we can gain a betterunderstanding of the asymptotic

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1.3Estimating Limit Values from Graphs

1.4Estimating Limit Values from Tables

1.5Determining Limits Using Algebraic Properties of Limits

1.6Determining Limits Using Algebraic Manipulation

1.7Selecting Procedures for Determining Limits

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