Section 2.2 The Limit of a Function
In computing tangent slopes, velocities and areas, there is a common step, new with calculus; calculating a limit, like calculating that as \(x\) approaches 1, \(\displaystyle\frac{x^2-1}{x-1}\) approaches 2:
\begin{equation*}
\lim_{x \to 1}\frac{x^{2}-1}{x-1} = 2.
\end{equation*}
We will now make this idea of limits more clear, and learn how to calculate them. With this key new skill, most other calculations in this course can be handled with familiar algebra, geometry, trigonometry and such.
Definition 2.2.1. Limit, informal version.
For a function \(f\) and numbers \(a\) and \(L\text{,}\) we say that the limit of \(f(x)\) as \(x\) approaches \(a\) is \(L\) if we can force the value of \(f(x)\) to be as close to \(L\) as we wish by considering only values of \(x\) sufficiently close to \(a\text{,}\) but not equal to \(a\text{.}\) This is written as
\begin{equation*}
\lim_{x \to a} f(x) = L
\end{equation*}
Note that the value of \(f(a)\) is irrelevant: \(f\) need not even be defined for \(x=a\text{.}\)
Example 2.2.2. A limit where the formula gives 0/0.
Guess the value that \(f(x)=\displaystyle\frac{x-1}{x^2-1}\) approaches as \(x\) approaches 1; that is, guess the value of \(\displaystyle\lim_{x \to 1}\frac{x-1}{x^2-1}\text{.}\)
Do this by trying \(x\) values that differ from 1 by \(0.1\text{,}\) then \(0.01\) and so on.
Then try to corroborate your guess by simplifying the formula for \(f(x)\text{.}\)
Example 2.2.3. Calculators can be fooled near 0/0.
Investigate the behavior of \(\displaystyle \frac{\sqrt{t^2+9}-3}{t^2}\) as \(t\) approaches 0.
First try \(x\) values \(\pm 1, \pm 0.5, \pm 0.1, \pm 0.05, \pm 0.01\text{,}\) and then the closer values \(\pm 0.0005, \pm 0.0001,\pm 0.00005, \pm 0.00001\) etc.
Warning: Things are not as they first seem here!
Solution.
The closer \(x\) values actually give a misleading result, due to the inability of a calculator to get sufficiently accurate results in this case. The limit is actually \(1/6\text{,}\) as suggested by the less close \(x\) values!
Using a graph on the calculator and zooming shows some probably unexpected and implausible behavior, revealing the accuracy limits of the calculator.
Example 2.2.4. \((\sin x) / x\) for \(x\) near 0.
How does the function
\(\displaystyle f(x)=\frac{\sin x}{x}\) behave as
\(x\) approaches 0? (See Example 2.4 in
OpenStax Calculus). 2
Setting \(x=0\) does not work as we get \(0/0\) again, but experiments on a graphing calculator suggest that the value approaches 1. (Note: remember to always use radian mode in math courses!)
This time, there is no simple algebraic way to simplify this formula and avoiding the “\(0/0\) problem”: we see in Chapter 3 how to compute this limit.
Example 2.2.5. \(\sin(\pi/x)\).
Explore how the function \(\displaystyle f(x)=\sin\frac{\pi}{x}\) behave as \(x\) approaches 0,
first with \(x=1,1/2,1/3,1/4,\dots,\)
then with \(x=2,2/3,2/5,2/7,2/9,\dots,\)
and finally with many more values, by using a graphing calculator and zooming in.
Solution.
Again we see that looking at only some nearby \(x\) values can be misleading. This time it seems that there is no one value that \(f(x)\) gets near to: no matter how close \(x\) is to 0, \(f(x)\) can be anywhere from \(-1\) to \(1\text{.}\)
This function has no limit as \(x \to 0\text{.}\)
Note well: Limits do not always exist! This example again shows that it is important to consider all values of \(x\) near \(a\) when studying a limit as \(x \to a\text{,}\) not just a selection.
Example 2.2.6. Measuring “closeness”.
Show that \(\lim_{x \to 1}f(x) = \lim_{x \to 1}\frac{2x^2-2}{x-1} = 4\text{.}\)
Solution.
We can simplify to \(f(x)=2x+2\text{,}\) valid for all \(x \neq 1\text{.}\)
Then we measure how close two numbers are by the absolute value of their difference. For example, if \(x\) is within \(0.001\) of \(1\text{,}\) \(|x-1| < 0.001\text{,}\) and so \(|f(x)-4| = |(2x+2)-4|=|2x-2|=2|x-1|\) which is less than 0.002.
When we look only at \(x\) values ever closer to 1, in that \(|x-1|\) is ever smaller, \(|f(x)-4=2|x-1|\) is ever smaller: \(f(x)\) gets ever closer to 4. For example, the value \(f(x)\) is sure to be within a tiny \(10^{-100}\) of 4 when we look at \(x\) values within \(0.5 \times 10^{-100}\) of \(1\text{.}\)
So the limit is 4.
Example 2.2.7. A function with a jump.
Consider the function \(f(x)\) given by
\begin{equation*}
f(x) = \left\{ \begin{array}{cl} \frac{2x^2-2}{x-1}, & x \neq 1 \\ 1, & x=1 \end{array} \right.
\end{equation*}
What is the limit of \(f(x)\) as \(x\) approaches 1?
Solution.Since the limit as
\(x \to a\) is based on values of
\(f(x)\) for all
\(x\) values
near to \(a\text{,}\) but
not equal to \(a\text{,}\) only the formula
\((2x^2-2)/(x-1)\) matters! And since it is equal to
\(2x+2\) for all
\(x\) values near
\(1\text{,}\) the value is near
\(2 \cdot 1+2=4\) there, and the limit is 4:
\(\lim_{x \to 1}f(x)=4\text{,}\) not 1.
Note well: The limit of \(f(x)\) as \(x\) goes to \(a\) does not always equal the value \(f(a)\text{,}\) even when \(f(a)\) makes sense!
The graph of this function has a
jump at
\(x=1\text{,}\) but the limit calculation ignores this, and treats the function as as if it were “uninterrupted” or “continuous” there.
Another type of jump: the Heaviside function.
In the physical description of sudden changes, like turning on a power switch, the Heaviside Function is often useful:
\begin{equation*}
H(t) = 0 \text{ for } t < 0; H(t) = 1 \text{ for } t \geq 0
\end{equation*}
For \(t\) near 0 and positive, \(H(t)\) is 1, suggesting a limit of 1. But for \(t\) near 0 and negative, \(H(t)\) is 0, suggesting a limit of 0.
The limit cannot be both zero and one, so again this function has no limit as \(t \to 0\), due to this jump from one value to another, which breaks the graph at this point.
One-sided Limits.
In the example above, we see that \(H(t)\) has “no limit” as \(t \to 0\text{,}\) but it is useful also to describe what happens at times just before \(t=0\text{,}\) and what happens at times just after \(t=0\text{:}\) what happens to one side or the other of a point on the graph.
We want to note that “as \(t\) approaches 0 from the right (\(t > 0 \)), \(H(t)\) approaches 1.” We use the notation \(t \to 0^+\text{,}\) with a plus sign superscript indicating that only \(t\) values to the right are considered: the relevant \(t\) values are “0 + something”.
The value approached is the right-hand limit, or the limit from the right, with short-hand notation
\begin{equation*}
\lim_{t \to 0^+}H(t) = 1.
\end{equation*}
One-sided limits: the left-hand limit.
Similarly the behavior for \(t\) near 0 and less than zero is called the left-hand limit and we use a minus sign superscript, because the \(t\) value is “0 minus something:”
\begin{equation*}
\lim_{t \to 0^-}H(t) = 0.
\end{equation*}
Note well: “\(t \to 0^-\)” is different from “\(t \to -0\)”, which would be a funny way of writing a normal “two-sided” limit. And \(t \to 1^-\) is very different than \(t \to -1\text{;}\) the former is about what happens for \(t\) just below 1; the latter is about what happens for \(t\) near \(-1\text{.}\)
Using one-sided limits to compute [two-sided] limits.
Sometime it is easier to compute each one-sided limit at \(a\) and then use these to learn about the regular “two-sided” limit:
Theorem 2.2.8.
If both one sided limits at \(a\) exist and are equal
\begin{equation*}
\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L
\end{equation*}
then that common value is also the limit there:
\begin{equation*}
\lim_{x \to a} f(x) = L
\end{equation*}
Otherwise, the latter limit does not exist.Infinite limits.
We have seen several ways that a function can fail to have a limit as \(x \to a\text{,}\) and decided that sometimes, there is still something useful to say about how the function behaves for \(x\) near \(a\text{.}\) Here is another case of that.
Example 2.2.9. Behavior of \(1/x^2\) as \(x \to 0\).
Investigate the behavior of \(f(x)=1/x^2\) as \(x\) approaches 0. Does it have a limit?
Solution.We see that there is no numerical value that
\(f(x)\) gets close to, but
there is a trend worth noting: The values of \(f(x)\) get larger and larger, with no upper bound.
Informally we could say that the value approaches infinity.
To describe cases like this, we introduce the symbol \(\infty\) for infinity and say that:
\begin{equation*}
\lim_{x \to 0} \frac{1}{x^2} = \infty
\end{equation*}
or in words, “as \(x\) approaches zero, the limit of \(1/x^2\) is infinity.”One-sided infinite limits.
Finally, it is natural to combine the ideas of infinite limits and one-sided limits.
Example 2.2.10.
Describe how \(f(x) = \frac{1}{x-2}\) behaves for \(x\) near 2, for the two cases \(x > 2\) and \(x < 2\)
Solution.
The values get large and positive on one side, large and negative on the other, so for \(x\) coming from the right, “the value approaches \(\infty\)”, while from the left, “the value approaches \(-\infty\)”.
Combining the above ideas and notation of one sided limits and infinite limits, we state this as
\begin{equation*}
\lim_{x \to 2^-}\frac{1}{x-2} = -\infty, \quad \lim_{x \to 2^+}\frac{1}{x-2} = \infty.
\end{equation*}
But the limits from the two sides are different, so
\begin{equation*}
\lim_{x \to 2}\frac{1}{x-2} \quad \mbox{Does Not Exist (DNE).}
\end{equation*}
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