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2.5: Limits at Infinity

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    Learning Objectives

    • Calculate the limit of a function as \(x\) increases or decreases without bound.
    • Recognize a horizontal asymptote on the graph of a function.

    In this section, we define limits at infinity and show how these limits affect the graph of a function.

    We begin by examining what it means for a function to have a finite limit at infinity. Then we study the idea of a function with an infinite limit at infinity.

    Limits at Infinity and Horizontal Asymptotes

    Recall that \(\displaystyle \lim_{x→a}f(x)=L\) means \(f(x)\) becomes arbitrarily close to \(L\) as long as \(x\) is sufficiently close to \(a\). We can extend this idea to limits at infinity. For example, consider the function \(f(x)=2+\frac{1}{x}\). As can be seen graphically in Figure \(\PageIndex{1}\) and numerically in Table \(\PageIndex{1}\), as the values of \(x\) get larger, the values of \(f(x)\) approach \(2\). We say the limit as \(x\) approaches \(∞\) of \(f(x)\) is \(2\) and write \(\displaystyle \lim_{x→∞}f(x)=2\). Similarly, for \(x<0\), as the values \(|x|\) get larger, the values of \(f(x)\) approach \(2\). We say the limit as \(x\) approaches \(−∞\) of \(f(x)\) is \(2\) and write \(\displaystyle \lim_{x→−∞}f(x)=2\).

    The function f(x) 2 + 1/x is graphed. The function starts negative near y = 2 but then decreases to −∞ near x = 0. The function then decreases from ∞ near x = 0 and gets nearer to y = 2 as x increases. There is a horizontal line denoting the asymptote y = 2.
    Figure \(\PageIndex{1}\):The function approaches the asymptote \(y=2\) as \(x\) approaches \(±∞\).
    Table \(\PageIndex{1}\): Values of a function \(f\) as \(x→±∞\)
    \(x\) 10 100 1,000 10,000
    \(2+\frac{1}{x}\) 2.1 2.01 2.001 2.0001
    \(x\) −10 −100 −1000 −10,000
    \(2+\frac{1}{x}\) 1.9 1.99 1.999 1.9999

    More generally, for any function \(f\), we say the limit as \(x→∞\) of \(f(x)\) is \(L\) if \(f(x)\) becomes arbitrarily close to \(L\) as long as \(x\) is sufficiently large. In that case, we write \(\displaystyle \lim_{x→∞}f(x)=L\). Similarly, we say the limit as \(x→−∞\) of \(f(x)\) is \(L\) if \(f(x)\) becomes arbitrarily close to \(L\) as long as \(x<0\) and \(|x|\) is sufficiently large. In that case, we write \(\displaystyle \lim_{x→−∞}f(x)=L\). We now look at the definition of a function having a limit at infinity.

    Definition: Limit at Infinity (Informal)

    If the values of \(f(x)\) become arbitrarily close to \(L\) as \(x\) becomes sufficiently large, we say the function \(f\) has a limit at infinity and write

    \[\lim_{x→∞}f(x)=L. \nonumber \]

    If the values of \(f(x)\) becomes arbitrarily close to \(L\) for \(x<0\) as \(|x|\) becomes sufficiently large, we say that the function \(f\) has a limit at negative infinity and write

    \[\lim_{x→−∞}f(x)=L. \nonumber \]

    If the values \(f(x)\) are getting arbitrarily close to some finite value \(L\) as \(x→∞\) or \(x→−∞\), the graph of \(f\) approaches the line \(y=L\). In that case, the line \(y=L\) is a horizontal asymptote of \(f\) (Figure \(\PageIndex{2}\)). For example, for the function \(f(x)=\dfrac{1}{x}\), since \(\displaystyle \lim_{x→∞}f(x)=0\), the line \(y=0\) is a horizontal asymptote of \(f(x)=\dfrac{1}{x}\).

    The figure is broken up into two figures labeled a and b. Figure a shows a function f(x) approaching but never touching a horizontal dashed line labeled L from above. Figure b shows a function f(x) approaching but never a horizontal dashed line labeled M from below.
    Figure \(\PageIndex{2}\): (a) As \(x→∞\), the values of \(f\) are getting arbitrarily close to \(L\). The line \(y=L\) is a horizontal asymptote of \(f\). (b) As \(x→−∞\), the values of \(f\) are getting arbitrarily close to \(M\). The line \(y=M\) is a horizontal asymptote of \(f\).
    Definition: Horizontal Asymptote

    If \(\displaystyle \lim_{x→∞}f(x)=L\) or \(\displaystyle \lim_{x→−∞}f(x)=L\), we say the line \(y=L\) is a horizontal asymptote of \(f\).

    A function cannot cross a vertical asymptote because the graph must approach infinity (or \( −∞\)) from at least one direction as \(x\) approaches the vertical asymptote. However, a function may cross a horizontal asymptote. In fact, a function may cross a horizontal asymptote an unlimited number of times. For example, the function \(f(x)=\dfrac{\cos x}{x}+1\) shown in Figure \(\PageIndex{3}\) intersects the horizontal asymptote \(y=1\) an infinite number of times as it oscillates around the asymptote with ever-decreasing amplitude.

    The function f(x) = (cos x)/x + 1 is shown. It decreases from (0, ∞) and then proceeds to oscillate around y = 1 with decreasing amplitude.
    Figure \(\PageIndex{3}\): The graph of \(f(x)=(\cos x)/x+1\) crosses its horizontal asymptote \(y=1\) an infinite number of times.

    The algebraic limit laws and squeeze theorem we introduced in Introduction to Limits also apply to limits at infinity. We illustrate how to use these laws to compute several limits at infinity.

    Example \(\PageIndex{1}\): Computing Limits at Infinity

    For each of the following functions \(f\), evaluate \(\displaystyle \lim_{x→∞}f(x)\) and \(\displaystyle \lim_{x→−∞}f(x)\). Determine the horizontal asymptote(s) for \(f\).

    1. \(f(x)=5−\dfrac{2}{x^2}\)
    2. \(f(x)=\dfrac{\sin x}{x}\)
    3. \(f(x)=\tan^{−1}(x)\)

    Solution

    a. Using the algebraic limit laws, we have

    \[\lim_{x→∞}\left(5−\frac{2}{x^2}\right)=\lim_{x→∞}5−2\left(\lim_{x→∞}\frac{1}{x}\right)\cdot\left(\lim_{x→∞}\frac{1}{x}\right)=5−2⋅0=5.\nonumber \]

    Similarly, \(\displaystyle \lim_{x→−∞}f(x)=5\). Therefore, \(f(x)=5-\dfrac{2}{x^2}\) has a horizontal asymptote of \(y=5\) and \(f\) approaches this horizontal asymptote as \(x→±∞\) as shown in the following graph.

    The function f(x) = 5 – 2/x2 is graphed. The function approaches the horizontal asymptote y = 5 as x approaches ±∞.
    Figure \(\PageIndex{4}\): This function approaches a horizontal asymptote as \(x→±∞.\)

    b. Since \(-1≤\sin x≤1\) for all \(x\), we have

    \[\frac{−1}{x}≤\frac{\sin x}{x}≤\frac{1}{x}\nonumber \]

    for all \(x≠0\). Also, since

    \(\displaystyle \lim_{x→∞}\frac{−1}{x}=0=\lim_{x→∞}\frac{1}{x}\),

    we can apply the squeeze theorem to conclude that

    \(\displaystyle \lim_{x→∞}\frac{\sin x}{x}=0.\)

    Similarly,

    \(\displaystyle \lim_{x→−∞}\frac{\sin x}{x}=0.\)

    Thus, \(f(x)=\dfrac{\sin x}{x}\) has a horizontal asymptote of \(y=0\) and \(f(x)\) approaches this horizontal asymptote as \(x→±∞\) as shown in the following graph.

    The function f(x) = (sin x)/x is shown. It has a global maximum at (0, 1) and then proceeds to oscillate around y = 0 with decreasing amplitude.
    Figure \(\PageIndex{5}\): This function crosses its horizontal asymptote multiple times.

    c. To evaluate \(\displaystyle \lim_{x→∞}\tan^{−1}(x)\) and \(\displaystyle \lim_{x→−∞}\tan^{−1}(x)\), we first consider the graph of \(y=\tan(x)\) over the interval \(\left(−\frac{π}{2},\frac{π}{2}\right)\) as shown in the following graph.

    The function f(x) = tan x is shown. It increases from (−π/2, −∞), passes through the origin, and then increases toward (π/2, ∞). There are vertical dashed lines marking x = ±π/2.
    Figure \(\PageIndex{6}\): The graph of \(y=\tan x\) has vertical asymptotes at \(x=±\frac{π}{2}\)

    Since

    \(\displaystyle \lim_{x→\tfrac{π}{2}^−}\tan x=∞,\)

    it follows that

    \(\displaystyle \lim_{x→∞}\tan^{−1}(x)=\frac{π}{2}.\)

    Similarly, since

    \(\displaystyle \lim_{x→-\tfrac{π}{2}^+}\tan x=−∞,\)

    it follows that

    \(\displaystyle \lim_{x→−∞}\tan^{−1}(x)=−\frac{π}{2}.\)

    As a result, \(y=\frac{π}{2}\) and \(y=−\frac{π}{2}\) are horizontal asymptotes of \(f(x)=\tan^{−1}(x)\) as shown in the following graph.

    The function f(x) = tan−1 x is shown. It increases from (−∞, −π/2), passes through the origin, and then increases toward (∞, π/2). There are horizontal dashed lines marking y = ±π/2.
    Figure \(\PageIndex{7}\): This function has two horizontal asymptotes.
    Exercise \(\PageIndex{1}\)

    Evaluate \(\displaystyle \lim_{x→−∞}\left(3+\frac{4}{x}\right)\) and \(\displaystyle \lim_{x→∞}\left(3+\dfrac{4}{x}\right)\). Determine the horizontal asymptotes of \(f(x)=3+\frac{4}{x},\) if any.

    Hint

    \(\displaystyle \lim_{x→±∞}\frac{1}{x}=0\)

    Answer

    Both limits are \(3.\) The line \(y=3\) is a horizontal asymptote.

    Infinite Limits at Infinity

    Sometimes the values of a function \(f\) become arbitrarily large as \(x→∞ \)(or as \(x→−∞\)). In this case, we write \(\displaystyle \lim_{x→∞}f(x)=∞\) (or \(\displaystyle \lim_{x→−∞}f(x)=∞\)). On the other hand, if the values of \(f\) are negative but become arbitrarily large in magnitude as \(x→∞\) (or as \(x→−∞\)), we write \(\displaystyle \lim_{x→∞}f(x)=−∞\) (or \(\displaystyle \lim_{x→−∞}f(x)=−∞\)).

    For example, consider the function \(f(x)=x^3\). As seen in Table \(\PageIndex{2}\) and Figure \(\PageIndex{8}\), as \(x→∞\) the values \(f(x)\) become arbitrarily large. Therefore, \(\displaystyle \lim_{x→∞}x^3=∞\). On the other hand, as \(x→−∞\), the values of \(f(x)=x^3\) are negative but become arbitrarily large in magnitude. Consequently, \(\displaystyle \lim_{x→−∞}x^3=−∞.\)

    Table \(\PageIndex{2}\): Values of a power function as \(x→±∞\)
    \(x\) 10 20 50 100 1000
    \(x^3\) 1000 8000 125,000 1,000,000 1,000,000,000
    \(x\) −10 −20 −50 −100 −1000
    \(x^3\) −1000 −8000 −125,000 −1,000,000 −1,000,000,000
    The function f(x) = x3 is graphed. It is apparent that this function rapidly approaches infinity as x approaches infinity.
    Figure \(\PageIndex{8}\): For this function, the functional values approach \(±\)infinity as \(x→±∞.\)
    Definition: Infinite Limit at Infinity (Informal)

    We say a function \(f\) has an infinite limit at infinity and write

    \[\lim_{x→∞}f(x)=∞. \nonumber \]

    if \(f(x)\) becomes arbitrarily large for \(x\) sufficiently large. We say a function has a negative infinite limit at infinity and write

    \[\lim_{x→∞}f(x)=−∞. \nonumber \]

    if \(f(x)<0\) and \(|f(x)|\) becomes arbitrarily large for \(x\) sufficiently large. Similarly, we can define infinite limits as \(x→−∞.\)

    Formal Definitions

    Earlier, we used the terms arbitrarily close, arbitrarily large, and sufficiently large to define limits at infinity informally. Although these terms provide accurate descriptions of limits at infinity, they are not precise mathematically. Here are more formal definitions of limits at infinity. We then look at how to use these definitions to prove results involving limits at infinity.

    Definition: Limit at Infinity (Formal)

    We say a function \(f\) has a limit at infinity, if there exists a real number \(L\) such that for all \(ε>0\), there exists \(N>0\) such that

    \[|f(x)−L|<ε \nonumber \]

    for all \(x>N.\) in that case, we write

    \[\lim_{x→∞}f(x)=L \nonumber \]

    The function f(x) is graphed, and it has a horizontal asymptote at L. L is marked on the y axis, as is L + ॉ and L – ॉ. On the x axis, N is marked as the value of x such that f(x) = L + ॉ.
    Figure \(\PageIndex{9}\): For a function with a limit at infinity, for all \(x>N, |f(x)−L|<ε.\)

    Earlier in this section, we used graphical evidence in Figure \(\PageIndex{1}\) and numerical evidence in Table \(\PageIndex{1}\) to conclude that \(\displaystyle \lim_{x→∞}\left(2+\frac{1}{x}\right)=2\). Here we use the formal definition of limit at infinity to prove this result rigorously.

    Example \(\PageIndex{2}\):

    Use the formal definition of limit at infinity to prove that \(\displaystyle \lim_{x→∞}\left(2+\frac{1}{x}\right)=2\).

    Solution

    Let \(ε>0.\) Let \(N=\frac{1}{ε}\). Therefore, for all \(x>N\), we have

    \[\left|2+\frac{1}{x}−2\right|=\left|\frac{1}{x}\right|=\frac{1}{x}<\frac{1}{N}=ε \nonumber \]

    Exercise \(\PageIndex{2}\)

    Use the formal definition of limit at infinity to prove that \(\displaystyle \lim_{x→∞}\left(3-\frac{1}{x^2}\right)=3\).

    Hint

    Let \(N=\frac{1}{\sqrt{ε}}\).

    Answer

    Let \(ε>0.\) Let \(N=\frac{1}{\sqrt{ε}}\). Therefore, for all \(x>N,\) we have

    \[\Big|3−\frac{1}{x^2}−3\Big|=\frac{1}{x^2}<\frac{1}{N^2}=ε \nonumber \]

    Therefore, \(\displaystyle \lim_{x→∞}(3−1/x^2)=3.\)

    We now turn our attention to a more precise definition for an infinite limit at infinity.

    Definition: Infinite Limit at Infinity (Formal)

    We say a function \(f\) has an infinite limit at infinity and write

    \(\displaystyle \lim_{x→∞}f(x)=∞\)

    if for all \(M>0,\) there exists an \(N>0\) such that

    \(f(x)>M\)

    for all \(x>N\) (see Figure \(\PageIndex{10}\)).

    We say a function has a negative infinite limit at infinity and write

    \(\displaystyle \lim_{x→∞}f(x)=−∞\)

    if for all \(M<0\), there exists an \(N>0\) such that

    \(f(x)<M\)

    for all \(x>N\).

    Similarly we can define limits as \(x→−∞.\)

    The function f(x) is graphed. It continues to increase rapidly after x = N, and f(N) = M.
    Figure \(\PageIndex{10}\): For a function with an infinite limit at infinity, for all \(x>N,\; f(x)>M.\)

    Earlier, we used graphical evidence (Figure \(\PageIndex{8}\)) and numerical evidence (Table \(\PageIndex{2}\)) to conclude that \(\displaystyle \lim_{x→∞}x^3=∞\). Here we use the formal definition of infinite limit at infinity to prove that result.

    Example \(\PageIndex{3}\)

    Use the formal definition of infinite limit at infinity to prove that \(\displaystyle \lim_{x→∞}x^3=∞.\)

    Solution

    Let \(M>0.\) Let \(N=\sqrt[3]{M}\). Then, for all \(x>N\), we have

    \(x^3>N^3=(\sqrt[3]{M})^3=M.\)

    Therefore, \(\displaystyle \lim_{x→∞}x^3=∞\).

    Exercise \(\PageIndex{3}\)

    Use the formal definition of infinite limit at infinity to prove that \(\displaystyle \lim_{x→∞}3x^2=∞.\)

    Hint

    Let \(N=\sqrt{\frac{M}{3}}\).

    Answer

    Let \(M>0.\) Let \(N=\sqrt{\frac{M}{3}}\). Then, for all \(x>N,\) we have

    \(3x^2>3N^2=3\left(\sqrt{\frac{M}{3}}\right)^2=\frac{3M}{3}=M\)

    Key Concepts

    • The limit of \(f(x)\) is \(L\) as \(x→∞\) (or as \(x→−∞)\) if the values \(f(x)\) become arbitrarily close to \(L\) as \(x\) becomes sufficiently large.
    • The limit of \(f(x)\) is \(∞\) as \(x→∞\) if \(f(x)\) becomes arbitrarily large as \(x\) becomes sufficiently large. The limit of \(f(x)\) is \(−∞\) as \(x→∞\) if \(f(x)<0\) and \(|f(x)|\) becomes arbitrarily large as \(x\) becomes sufficiently large. We can define the limit of \(f(x)\) as \(x\) approaches \(−∞\) similarly.

    Glossary

    horizontal asymptote
    if \(\displaystyle \lim_{x→∞}f(x)=L\) or \(\displaystyle \lim_{x→−∞}f(x)=L\), then \(y=L\) is a horizontal asymptote of \(f\)
    infinite limit at infinity
    a function that becomes arbitrarily large as \(x\) becomes large
    limit at infinity
    a function that approaches a limit value \(L\) as \(x\) becomes large

    Contributors and Attributions

    • Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.


    This page titled 2.5: Limits at Infinity is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.