Limits at Infinity and Asymptotes

Infinity is an intriguing concept to explore, especially looking at what happens to functions as values approach infinity, i.e., limits at infinity. While exploring limits at infinity, you will encounter another mathematical concept: asymptotes. No, 'asymptotes' are not another type of tote bag, like the one you find at your school or university's gift shop. This strange-sounding term 'asymptote', at least in mathematics, is a straight line that accompanies certain functions and has some intriguing properties. Let's take a look.

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    Limits at Infinity and Asymptotes meaning

    So, what happens to different functions as they approach infinity? Well, as a function approaches infinity, it gets closer to its limit at infinity.

    If a function \(y=f(x)\) is given, the limit of this function at infinity is not simply \(\infty\) itself. The word 'limit' is crucial here. When finding the limit of a function, you are not calculating the value of the function at infinity, which is mathematical nonsense, as infinity is not a number.

    It is all about knowing what happens to the function \(f(x)\) as the value of \(x\) approaches infinity. There are two potential outcomes as the input to a function approaches infinity: either the limit diverges to infinity or converges to a certain value.

    Limits at Infinity and Asymptotes limit of a function reaching a value StudySmarterFigure 1.- Limit of a function approaching a certain value at infinity

    From the above diagram, you can see that as \(x \rightarrow \infty\), the function \(f(x)\) reaches a constant, which is bounded by the line \(y=l\). We write it as:

    $$\lim_{x \to \infty} f(x)= l$$

    Hence, the limit of the function is \(l\).

    In other words, the function converges to \(l\) but never reaches it; it only gets closer and closer.

    Some typical examples are the exponential and the logarithmic functions. The straight line that bounds these values is known as an asymptote. An asymptote is rigorously defined as follows:

    A line that constantly approaches a curve and virtually bounds it but never actually meets it, is known as an asymptote.

    In simpler terms, an asymptote is an imaginary line that a curve approaches as its input tends to infinite. Remember that the asymptote and the curve never actually meet, but at infinite, they are infinitely close.

    There are three types of asymptotes:


    Horizontal Asymptotes
  • Vertical Asymptotes

  • Oblique or Slant Asymptotes.


  • Connection between limits at infinity and Horizontal Asymptotes

    As the name suggests, horizontal asymptotes are horizontal, i.e. parallel to the \(x\)-axis. The slope of any horizontal line is \(0\). Let's take a look at the example of a decreasing function again:

    Limits at Infinity and Asymptotes Horizontal asymptote to a curve StudySmarterFigure 2.- The Horizontal Asymptote to a curve defined by \(y=f(x)\)

    Notice that the limit of the function converges to a finite value as \(x \rightarrow \infty\), which gives rise to the horizontal asymptote of the function.

    It can be seen that the asymptote (the straight line) never crosses or even meets the curve itself. It may look like they meet each other at some point, but they only get ever closer.

    As the value of \(x\) tends to \(+\infty\) the function tends to a certain value. As \(x\) approaches \(-\infty\), the value of the function starts to blow up, it approaches \(+\infty\).

    For a curve described by the function \(y=f(x)\), if the function converges to a constant value, \(b\), then the equation of the horizontal asymptote is given by \(y=b\). In other words, \(y=f(x) \rightarrow b\) as \(x \rightarrow +\infty\).

    The symbol '\(\rightarrow\)' denotes 'approaches', as a phrase, it can be read as 'as \(x\) approaches \(+\infty\)'.

    The correct way to denote this is with limit notation.

    \[\lim_{x \to +\infty} f(x)=b \]

    All this equation means is that the limit of the function \(f(x)\) as \(x\) tends to infinite is \(b\).

    Now, from this statement \(b\) could be an asymptote of \(f(x)\), but it might not. So, how can you find a horizontal asymptote?

    How to find the Horizontal asymptotes of a function?

    There is no universal algorithm or method for finding horizontal asymptotes, but there are some rules you can use to identify horizontal asymptotes of rational functions. If a rational function consists of a polynomial in both the numerator and the denominator, then you can find the asymptotes according to the following steps:

    Keep in mind that the degree of a polynomial is defined as the highest power of the variable.

    • If the degree of the polynomial in the numerator is greater than the degree of the polynomial in the denominator, then no horizontal asymptote exists for that curve.

    • If the degree of the polynomial in the numerator is lower than the degree of the polynomial in the denominator, then the equation of the horizontal is always \(y=0\).

    • If the degree of the numerator is equal to the degree of the denominator, then divide by the leading coefficient (the number multiplied by the variable of the highest degree) of the numerator by the leading coefficient of the denominator. The quotient of these leading coefficients is the y value of the horizontal asymptote.

    Recall that a Rational Function is defined as a function that can be expressed as a ratio of two constituent functions, i.e., \(R(x)=\frac{p(x)}{q(x)}\).

    Find the horizontal asymptote in the function \( \displaystyle y=\frac{3x-2}{4x}\).

    Solution:

    It can be observed that the degree of the polynomial in the numerator and denominator is the same.

    Therefore, to find the horizontal asymptote, simply divide the leading coefficient of the numerator by the leading coefficient of the denominator.

    \[y = \frac{3}{4}\]

    Vertical asymptotes and limit at infinity calculation

    Vertical asymptotes are asymptotes which are parallel to the y-axis. In this case, for a finite value of \(x\), the function is undefined. Let's look at another example in the function below. Instead of the independent variable, the dependent variable attains the value of \(+\infty\). It can also tend to \(-\infty\) for a finite value of \(x\).

    Limits at Infinity and Asymptotes Vertical asymptote to a curve StudySmarterFigure 3.- The Vertical Asymptote to a curve defined by \(y=f(x)\)

    Here, the curve is approaching \(x=a\), but never touches it. Therefore, the vertical line \(x=a\) is a vertical asymptote. And again, if the value tends to something and never actually reaches it, it can be mathematically expressed as a limit that diverges:

    $$ \lim_{x \to a} f(x)= +\infty \ \ \text{or} \ \ \lim_{x \to a} f(x)=-\infty$$

    where the vertical asymptote is given by \(x=a\).

    How to find Vertical Asymptotes of a function?

    You saw how a vertical asymptote is defined mathematically, the function needs to approach some value \(x=a\) so that \(y\) approaches \(+\infty\) or \(-\infty\). You need to have such a value of \(a\) that the limits become undefined \((+\infty \ \text{or} \ -\infty)\).

    Have you seen examples of this in your time studying math? You have! Dividing by zero is a huge no-no in math!

    Thus, for every rational function, you must find a value of \(x\) such that the denominator will become \(0\). For instance,

    \[y = \frac{5x}{x-3}.\]

    is undefined when \(x=3\).

    Some functions never equate to \(0\) for any real value of \(x\), for example, a class of exponential functions never attain the value \(0\). This implies that those functions don't have any vertical asymptotes. Let's look at an example to see what the above process looks like in practice:

    Find the vertical asymptotes for the function \(y= \displaystyle \frac{x+6}{2x+4}\).

    Solution:

    As discussed earlier, equate the denominator to \(0\) and solve for \(x\):

    $$ \begin{aligned} 2x+4 &=0 \\ x &=-2 \end{aligned} $$

    So as \(x \rightarrow -2\) the denominator also tends to \(0\). As the function is defined over all real numbers, we don’t need to calculate the \( \text{LHL}\)(Left Hand Limit) and \(RHL\)(Right Hand Limit) separately. Calculating the limit as \(x \rightarrow -2\) we get:

    $$ \lim_{x \to -2} \frac{x+6}{2x+4}=\frac{-2+6}{0}=+\infty$$

    Hence, the vertical asymptote for the function \(y=\frac{x+6}{2x4}\) is \(x=-2\).

    Relationship between limits at infinity and Slant asymptotes

    Asymptotes don't always have to be exactly horizontal or vertical, they can be in any orientation. Asymptotes that make an acute angle with the x-axis are known as slant asymptotes.

    Limits at Infinity and Asymptotes Slant asymptote to a curve StudySmarterFigure 4.- The Slant Asymptote to a curve defined by \(y=f(x)\)

    The notion is very similar to that of the horizontal asymptotes, since one has to consider the limit of the function at infinity.

    If you consider the limit of the function as \(x \rightarrow \infty\), the curve converges to a certain value which can be described by a straight line, which is the Slant Asymptote to the curve.

    Let a function define a curve by \(y=f(x)\), then the slant asymptote to the curve is given by \(y=mx+b\), if and only if the below limits are satisfied:

    $$ \lim_{x \to +\infty} \frac{f(x)}{x}=m \ \text{and} \ \lim_{x \to +\infty}|f(x)-mx|=b$$

    If the above limits are not finite, then there are no slant asymptotes to the curve.

    If one compares all the above three types of asymptotes, it can be observed that vertical and horizontal asymptotes are only specific cases of slant asymptotes.

    Prove that the curve defined by the function \(f(x)= \displaystyle \frac{x-2}{2x+1}\) has no slant asymptotes.

    Solution:

    Calculating the limit as \(x \rightarrow +\infty\), we get the following limit,

    $$ \lim_{x \to +\infty} \frac{f(x)}{x}=0$$

    This yields \(m=0\) and thus the asymptote assumes the form \(y=b\), which is a horizontal asymptote and not a slant asymptote.

    Limits at Infinity and Asymptotes Examples

    (1)

    Find the horizontal asymptotes to the curve defined by the function \[y=\displaystyle \frac{2x}{-x^2+x+3}.\]

    Solution:

    For the numerator, observe that the highest power of \(x\) is \(1\) and so the degree of the numerator is \(1\).

    Now, for the denominator, it can be seen that the highest power of \(x\) is \(2\), so the degree is \(2\).

    Since the degree of the numerator is smaller than the degree of the denominator, the horizontal asymptote of this curve is:

    $$y=0$$

    (2)

    Find the horizontal asymptote to the curve defined by the function \[y= \displaystyle \frac{7x^2-23}{3x+5}.\]

    Solution:

    Observe that the highest power of \(x\) in the numerator is \(2\) and so the degree is \(2\).

    For the denominator, the highest power of \(x\) is \(1\) and so the degree is \(1\).

    And since \(2>1\), the degree of the numerator is greater than that of the denominator.

    Hence, there exist no horizontal asymptotes to this curve.

    (3)

    Find the vertical asymptote(s) to the curve defined by the function

    \[f(x) = \frac{5x}{x^2-5x+6}.\]

    Solution:

    Finding the vertical asymptote is a simple case of equating the denominator of the function to \(0\) and solving for \(x\).

    \[\begin{align} 0 &= x^2 - 5x + 6 \\ 0&= (x-2)(x-3) \\ 0&=x-2 \\ x&= 2 \\ 0&= x-3 \\ x&=3 \end{align}\]

    The vertical asymptotes are therefore

    \[x=2 \text{ and } x=3.\]

    Limits at Infinity and Asymptotes - Key takeaways

    • If the limit of a function reaches a finite value as \(x \rightarrow \infty\) then the limit of the function converges.

    • Asymptotes are lines on a graph that the curve gets really close to, which bound a curve in such a way that as the curve stretches to infinity, but never actually reaches it.

    • There are three types of asymptotes: Horizontal Asymptotes, Vertical Asymptotes, and Slant Asymptotes.

    • Horizontal asymptotes are parallel to the x-axis and their slope is \(0\), and as \(x \rightarrow \infty \) or \(x \rightarrow -\infty\), the value \(y\) reaches is the equation of the horizontal asymptote.

    • Vertical Asymptotes are parallel to the y-axis and their slope is undefined. If the following limit is satisfied, then \(x=a\) is a vertical asymptote to the curve: \( \lim_{x \to a} f(x) = \infty \ \text{or} \ -\infty\) .

    • Slant asymptotes are asymptotes that make an acute angle with the x-axis. A curve has a slant asymptote in the form \(y=mx+b\). They occur when the degree of the numerator is greater than the degree of the denominator, if and only if the following limits are finite: \( \displaystyle \lim_{x \to \infty} \frac{f(x)}{x}=m\) and \( \displaystyle \lim_{x \to \infty} |f(x)-mx|=b\).

    Frequently Asked Questions about Limits at Infinity and Asymptotes

    How do you find asymptotes with limits at infinity?

    By observing what number they get close as x grows.

    How do horizontal asymptotes relate to limits at infinity?

    The horizontal asymptotes of a function define the value of the limit as the function grows to infinity.

    Can there be an asymptote at infinity?

    Yes, some complex functions composed of fractions have limits to infinity.

    What is the limit when there is an asymptote?

    When there is an asymptote the limit of the function is equal to the value of the asymptote. For example if there is an horizontal asymptote equal to y=2 when f(x) goes to infinity, then the limit of the function at infinity is y=2.

    How do you find limits at infinity examples?

    To find a limit to the infinity, you need to find the value at which the function gets closer as x increases its value.

    An easy example is the function f(x)=1/x. Here you will need to increase the value of x.

    x=1, y=1
    x=10, y=0.1
    x=100, y=0.01
    x=1000, y=0.001

    As you can see, as x grows, y gets closer to zero. The limit in this example when x goes to infinity is y=0.

    Test your knowledge with multiple choice flashcards

    How many types of asymptotes can a rational function have?

    If a function \(y=f(x)\) reaches a finite value as \(x \rightarrow \infty\), does the limit \(\lim_{x \rightarrow \infty} f(x)\) converge or diverge? 

    The limit is called one sided if \(x\) approaches \(a\) from just two directions. 

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