Up until this point we have discussed what happens to a function as we move its input \(x\) closer and closer to a particular point \(a\text{.}\) For a great many applications of limits we need to understand what happens to a function when its input becomes extremely large — for example what happens to a population at a time far in the future.
The definition of a limit at infinity has a similar flavour to the definition of limits at finite points that we saw above, but the details are a little different. We also need to distinguish between positive and negative infinity. As \(x\) becomes very large and positive it moves off towards \(+\infty\) but when it becomes very large and negative it moves off towards \(-\infty\text{.}\)
The dotted horizontal lines indicate the behaviour as \(x\) becomes very large. The function on the left has limits as \(x \to \infty\) and as \(x \to -\infty\) since the function “settles down” to a particular value. On the other hand, the function on the right does not have a limit as \(x \to -\infty\) since the function just keeps getting bigger and bigger.
Note that, as was the case in Theorem 1.4.9, we need a little extra care with powers of functions. We must avoid taking square roots of negative numbers, or indeed any even root of a negative number 1
To be more precise, there is no real number \(x\) so that \(x^\text{even power}\) is a negative number. Hence we cannot take the even-root of a negative number and express it as a real number. This is precisely what complex numbers allow us to do, but alas there is not space in the course for us to explore them.
This is only true if the denominator of \(r\) is not an even number 2
where we write \(r = \frac{p}{q}\) with \(p,q\) integers with no common factors. For example, \(r = \frac{6}{14}\) should be written as \(r = \frac{3}{7}\) when considering this rule.
\(\ds \lim_{x \to \infty} \frac{1}{x^{1/2}} = 0\text{,}\) but \(\ds \lim_{x \to -\infty} \frac{1}{x^{1/2}}\) does not exist, because \(x^{1/2}\) is not defined for \(x \lt 0\text{.}\)
Our first application of limits at infinity will be to examine the behaviour of a rational function for very large \(x\text{.}\) To do this we use a “trick”.
As \(x\) becomes very large, it is the \(x^2\) term that will dominate in both the numerator and denominator and the other bits become irrelevant. That is, for very large \(x\text{,}\)\(x^2\) is much much larger than \(x\) or any constant. So we pull out these dominant parts
\begin{align*}
\frac{x^2-3x+4}{3x^2+8x+1}
&= \frac{x^2 \left(1-\frac{3}{x}+\frac{4}{x^2}\right)}
{x^2 \left(3+\frac{8}{x}+\frac{1}{x^2} \right)}\\
&= \frac{1-\frac{3}{x}+\frac{4}{x^2}}
{3+\frac{8}{x}+\frac{1}{x^2}} & \text{ remove the common factors}
\end{align*}
The biggest contribution to the numerator comes from the \(4x^2\) inside the square-root. When we pull \(x^2\) outside the square-root it becomes \(x\text{,}\) so the numerator is dominated by \(x \cdot \sqrt{4} = 2x\)
Now let us also think about the limit of the same function, \(\frac{\sqrt{4x^2+1}}{5x-1}\text{,}\) as \(x \rightarrow -\infty\text{.}\) There is something subtle going on because of the square-root. First consider the function 3
Just to change things up let’s use \(t\) and \(h(t)\) instead of the ubiquitous \(x\) and \(f(x)\text{.}\)
We’ll get much the same thing for any \(t \geq 0\text{.}\) For any \(t \ge 0\text{,}\)\(h(t)=\sqrt{t^2}\) returns exactly \(t\text{.}\) However now consider the function at \(t=-3\)
This is because when we defined \(\sqrt{\text{ }}\text{,}\) we defined it to be the positive square-root. i.e. the function \(\sqrt{t}\) can never return a negative number. So being more careful
\begin{align*}
h(t) &= \sqrt{t^2} = | t |
\end{align*}
Where the \(|t|\) is the absolute value of \(t\text{.}\) You are perhaps used to thinking of absolute value as “remove the minus sign”, but this is not quite correct. Let’s sketch the function
We use the same trick — try to work out what is the biggest term in the numerator and denominator and pull it to one side. Since we are taking the limit as \(x \to -\infty\) we should think of \(x\) as a large negative number.
The biggest contribution to the numerator comes from the \(4x^2\) inside the square-root. When we pull the \(x^2\) outside a square-root it becomes \(|x| = -x\) (since we are taking the limit as \(x \to -\infty\)), so the numerator is dominated by \(-x\cdot\sqrt{4} = -2x\)
So the limit as \(x \to -\infty\) is almost the same but we gain a minus sign. This is definitely not the case in general — you have to think about each example separately.
When \(x\) is very large, \(x^{7/5} = x\cdot x^{2/5}\) will be much larger than \(x\text{,}\) so the \(x^{7/5}\) term will dominate the \(x\) term. So factor out \(x^{7/5}\) and rewrite it as
Consider what happens to each of the factors as \(x \to \infty\)
For large \(x\text{,}\)\(x^{7/5} \gt x\) (this is actually true for any \(x \gt 1\)). In the limit as \(x \to +\infty\text{,}\)\(x\) becomes arbitrarily large and positive, and \(x^{7/5}\) must be bigger still, so it follows that
In the previous section we saw that finite limits and arithmetic interact very nicely (see Theorems 1.4.3 and 1.4.9). This enabled us to compute the limits of more complicated function in terms of simpler ones. When limits of functions go to plus or minus infinity we are quite a bit more restricted in what we can deduce. The next theorem states some results concerning the sum, difference, ratio and product of infinite limits — unfortunately in many cases we cannot make general statements and the results will depend on the details of the problem at hand.
Note that by “undetermined” we mean that the limit may or may not exist, but cannot be determined from the information given in the theorem. See Example 1.4.7 for an example of what we mean by “undetermined”. Additionally consider the following example.
Say we want to compute the limit of the difference of two of the above functions as \(x \to 0\text{.}\) Then the previous theorem cannot help us. This is not because it is too weak, rather it is because the difference of two infinite limits can be, either plus infinity, minus infinity or some finite number depending on the details of the problem. For example,
Give a polynomial \(f(x)\) with the property that both \(\displaystyle\lim_{x \rightarrow\infty} f(x)\) and \(\displaystyle\lim_{x \rightarrow -\infty} f(x)\) are (finite) real numbers.
Give a rational function \(f(x)\) with the properties that \(\displaystyle\lim_{x \rightarrow\infty} f(x) \neq \displaystyle\lim_{x \rightarrow -\infty} f(x)\text{,}\) and both limits are (finite) real numbers.
Suppose the concentration of a substance in your body \(t\) hours after injection is given by some formula \(c(t)\text{,}\) and \(\displaystyle\lim_{t \rightarrow \infty} c(t) \neq 0\text{.}\) What kind of substance might have been injected?
