Infinite Limits & Asymptotes

Introduction 1.2.1  
We have established that (1.1.17) doesn't make much sense, but when taking $L=\infty$, it would already seem a little better. Therefore, in this section, we will investigate what it means if the limit of a function in a point $c$ happens to be infinite, i.e. we will give formal definitions of

$$\lim_{x \to c} f(x) = +\infty \hbox{\ \ and\ \ }\lim_{x \to c} f(x) = -\infty$$ (1.2.1)

In applications, one is often interested in the asymptotic behaviour of a function, i.e. in what the function does in the long run. Intuitively, this is represented by

$$\lim_{x \to +\infty} f(x) =L$$ (1.2.2)

We will also establish a formal definition of (1.2.2) and see how to calculate it in a few cases.
It will turn out that both (1.2.1) and (1.2.2) will turn out to represent the asymptotes of a function.

Definitions 1.2.2  

• The limit of $f$ for $x$ going to $c$ is plus infinity, written as

  $$\lim_{x \to c} f(x) = +\infty \hbox{\ \ iff \ \ } \forall N > 0, \exists \delta > 0 \hbox{ such that } \vert x-c\vert < \delta \implies f(x) > N$$ (1.2.3)

  
  
  This means that $f(x)$ can be arbitrarily large, as long as you choose $x$ close enough to $c$.
• The limit of $f$ for $x$ going to $c$ is minus infinity, written as

  $$\lim_{x \to c} f(x) = -\infty \hbox{\ \ iff \ \ } \forall N > 0, \exists \delta > 0 \hbox{ such that } \vert x-c\vert < \delta \implies f(x) < -N$$ (1.2.4)

  
  
  This means that $f(x)$ can be negative and arbitrarily large in absolute value, as long as you choose $x$ close enough to $c$.

Definition 1.2.3  
The line $x=c$ is a vertical asymptote of $f$ iff

$$\lim_{x \to c^-} f(x) = \pm\infty \hbox{\ \ \ or \ \ } \lim_{x \to c^+} f(x) = \pm\infty$$ (1.2.5)

The notation $\pm$ signifies that it can be either $+\infty$ or $-\infty$. So if one of these limits happens to be $+\infty$ or $-\infty$, then we have a vertical asymptote.

Example 1.2.4  
An example where the left limit and the right limit are equal is

$$f: \hbox {I \hskip -5.2pt {R}}-\{5\} \to \hbox {I \hskip -5.2pt {R}}: x \mapsto x^2 + \frac{1}{(x-5)^2}$$ (1.2.6)

This function has a vertical asymptote, namely $x=5$ as is very clear from the graph:

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Written in this way, it is rather obvious that $f(x)$ is essentially a parabola with a perturbation that doesn't contribute all that much once you're far enough away from $5$. Usually $f$ will ``disguise" itself as

$$f(x)=\frac{x^4-10x^3+25x^2+1}{x^2-10x+25}$$ (1.2.7)

Of course, with a little elementary algebra, $f$ can be rewritten in the more instructive form (1.2.6).

Example 1.2.5  
An example where the left limit and the right limit are not equal (and therefore the limit does not exist) is

$$f: \hbox {I \hskip -5.2pt {R}}-\{5\} \to \hbox {I \hskip -5.2pt {R}}: x \mapsto x^2 + \frac{1}{x-5}$$ (1.2.8)

Also this function has the asymptote $x=5$ as is quite clear from the graph:

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Written in this way, it is rather obvious that $f(x)$ is essentially a parabola with a perturbation that doesn't change much once you are away from $5$. Usually $f$ will ``disguise" itself as

$$f(x)=\frac{x^3-5x^2+1}{x-5}$$ (1.2.9)

Of course, with a little elementary algebra, $f$ can be rewritten in the more instructive form (1.2.8).

Remark 1.2.6  
Note that a function can have a multitude of vertical asymptotes. In fact even an infinite number, like e.g. the tangent function.

Definitions 1.2.7  

• The limit of $f$ for $x$ going to plus infinity equals $L$, written as

  $$\lim_{x \to +\infty} f(x) = L \hbox{\ \ iff\ \ } \forall \varepsi... ...sts x_0 \hbox{ such that } \forall x > x_0 : \vert f(x) - L \vert < \varepsilon$$ (1.2.10)

  
  

• The limit of $f$ for $x$ going to minus infinity equals $L$, written as

  $$\lim_{x \to -\infty} f(x) = L \hbox{\ \ iff\ \ } \forall \varepsi... ...sts x_0 \hbox{ such that } \forall x < x_0 : \vert f(x) - L \vert < \varepsilon$$ (1.2.11)

  
  

Remark 1.2.8 (Horizontal Assymptotes)  
Saying that $$\lim_{x \to +\infty} f(x) =L$$ is the same as saying that

$$\lim_{x \to +\infty} f(x) - g(x) = 0$$ (1.2.12)

where $g(x)=L$, i.e. the equation of a horizontal line. Essentially (1.2.12) signifies that as $x$ gets bigger $f$ and $g$ approach each other, i.e. $f$ approaches a horizontal line. Such a horizontal line is called a horizontal asymptote.
The same reasoning holds for

$$\lim_{x \to -\infty} f(x) - g(x) = 0$$ (1.2.13)

so a function can have two horizontal asymptotes.

Definition 1.2.9  
The line $y=L$ is a horizontal asymptote of $f$ iff

$$\lim_{x \to +\infty} f(x) = L \hbox{\ \ or \ \ } \lim_{x \to -\infty} f(x) = L$$ (1.2.14)

Note that as only the limit to $+\infty$ and the limit to $-\infty$ are taken into consideration, so a function can have at most $2$ horizontal asymptotes.

Example 1.2.10  
An example of a function that has two horizontal asymptotes is

$$f : \hbox {I \hskip -5.2pt {R}}\to \hbox {I \hskip -5.2pt {R}}: x \mapsto \arctan(x)$$ (1.2.15)

This function has horizontal asymptotes $y=\frac{\pi}{2}$ and $y=-\frac{\pi}{2}$. The graph looks like:

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Remark 1.2.11  
Reading about vertical and horizontal asymptotes, one wonders if there are other ones. After all, there are other lines than just vertical and horizontal lines. Take a look at the function

$$f: \hbox {I \hskip -5.2pt {R}}\to \hbox {I \hskip -5.2pt {R}}: x \mapsto x+10xe^{-x^2}$$ (1.2.16)

where the factor $10$ is there to have a more noticeable effect on the graph.

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The graph seems to suggest that $y=x$ is an asymptote for the function $f$.

Definition 1.2.12  
The line $y=mx+q$ is a slanted asymptote of $f$ iff

$$\lim_{x \to +\infty} f(x) - (mx+q) = 0 \hbox{\ \ or \ \ } \lim_{x \to -\infty} f(x) - (mx+q) = 0$$ (1.2.17)

where $m$ and $q$ are two real numbers to be determined when calculating the limit.

Remark 1.2.13  
Note that if in (1.2.17), $m$ turns out to be 0, the asymptote found is a horizontal asymptote. This implies the following:

$$\char93 \hbox{horizontal asymptotes} + \char93 \hbox{slanted asymptotes} \leq 2$$ (1.2.18)

Example 1.2.14  
When searching the asymptotes of the function

$$f: \hbox {I \hskip -5.2pt {R}}\to \hbox {I \hskip -5.2pt {R}}: x \mapsto \frac{2x^2-x-14}{x-3}$$ (1.2.19)

The first thing to notice is that the denominator is 0 if $x=3$. So there could very well be a vertical asymptote. Indeed,

$$\lim_{x \to 3^-} \frac{2x^2-x-14}{x-3} = \frac{1}{0^-} = -\infty ... ...x{\ \ and\ \ } \lim_{x \to 3^+} \frac{2x^2-x-14}{x-3} = \frac{1}{0^+} = +\infty$$ (1.2.20)

So we have the vertical asymptote $x=3$. Note that calculating one of the limits suffices to determine the existence of the asymptote. To get a better idea of the graph of the function, both are useful. To see if there is a slanted asymptote, notice that

$$\lim_{x \to +\infty} \frac{2x^2-x-14}{x-3} - (mx+q) = \lim_{x \to +\infty} \frac{2x^2-x-14-(mx+q)(x-3)}{x-3} \cr$$ (1.2.21)

In order for this limit to be 0, we need $m=2$ to get rid off the first term, and consequently $q=5$ to get rid of the second term. The last term will be 0 no matter what number $q$ is. The limit to $-\infty$ will lead to exactly the same calculations.
A slick way of avoiding the unpleasant algebra in (1.2.21) is to notice (e.g. by doing the long division) that

$$\frac{2x^2-x-14}{x-3} = 2x+5 + \frac{1}{x-3}$$ (1.2.22)

and then it becomes obvious that when $x$ goes to infinity $f(x)$ will approach $2x+5$. Again, the graph looks like one would expect from the above calculations:

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