Like beads on a string leading to a pendant, points on a graph can lead you to the limit of a function. How can we use points on a graph to evaluate limits? Good question! Here we look at some of the different ways of finding limits of functions!
Finding Limits in Calculus
There are lots of ways to find the limit of a function!
You can use the \(\epsilon\), \(\delta\) definition of the limit and write a proof. See Limits of a Function for examples of this technique.
You can look at the graph or a table of values to see what the limit might be. See Finding Limits Using a Graph or Table for plenty of examples of finding limits this way.
You could look at the limit from the left and the limit from the right of a function, and see if they are the same. See One-Sided Limits for definitions and examples of using this technique.
You could use Limit Laws, which are theorems that have already been proven to find the limit. If your function is nice this is often the way people find the limit. For more information on properties of limits see Limit Laws
You may need to use a special theorem to find the limit, like the Squeeze Theorem or the Intermediate Value Theorem. Both of them are very useful and the Intermediate Value Theorem will turn up later in topics like finding the maximum value of a function. See The Squeeze Theorem or see The Intermediate Value Theorem for details on how to use them.
Here you will see a sampling of the ways to find the limit of a function.
Using the Definition of the Limit
To review the definition of the limit of a function, see Limits of a Function.
Take \(f(x)=k\) where \(a\) and \(k\) are constant real numbers. Is it true that
\[lim_{x \rightarrow a} f(x)=k\]
Answer:
Yes. Using the definition, for any \(\epsilon > 0\) you are given,
\[|f(x)-k|=|k-k|=0< \epsilon\]
no matter what \(\delta\) you use. So constant functions have the limit you would expect them to.
Take \(f(x)=x\), and let \(a\) be a constant real number. How do you know that
\[lim_{x \rightarrow a} f(x)=a\]
Answer:
You might be tempted to say "of course, the limit is \(a\) - the function is just a line". In fact, that is almost enough. You can't use any of the Properties of Limits, but you can use the definition and take \(\delta = \epsilon\) to show that the limit is \(a\).
Using the Rules for Finding Limits
For a review of the various properties of limits and how to use them , see Limit Laws.
Take the function \(f(x)=2x^3-3x^2+7\), and \(a\) to be a constant real number. Find
\[lim_{x \rightarrow a} f(x)\]
Answer:
Notice that the function is just the sum and product of powers of \(x\) along with the constant \(7\). You already know that
\(lim_{x \rightarrow a} x=a\) and \(lim_{x \rightarrow a} 7 =7\)
from the two examples above, which means the conditions to apply the Sum Rule, Product Rule, and Constant Rule are met. Then applying them gives
\[lim_{x \rightarrow a} f(x)= lim_{x \rightarrow } ( 2x^3-3x^2+7)\]
\[lim_{x \rightarrow a} f(x)=2a^3-3a^2+7\]
Finding Limits Graphically
Below is an example of using the graph to find the limit of a function. For more information on problems like this see Finding Limits Using a Graph or Table.
Consider the function
\[f(x)=\dfrac{1}{4}(x+1)(x-1)(x-5)\]
Find the limit of the function as \(x \rightarrow 3 \).
Answer:
First, graph the function and make a table of values near \(x=3\). Although the function has more roots than are shown in the graph, since you only care about the limit as \(x \rightarrow 3\), it makes sense to zoom in on the function there.
Using a graph with multiple points to find the limit of a function in red.
| \(x\) | \(f(x)\) |
| 2.5 | -3.28 |
| 2.55 | -3.37 |
| 2.6 | -3.46 |
| 2.65 | -3.54 |
| 2.7 | -3.62 |
| 2.75 | -3.69 |
| 2.8 | -3.76 |
| 2.85 | -3.83 |
| 2.9 | -3.89 |
| 2.95 | -3.95 |
| 3.0 | -4.0 |
| 3.05 | -4.05 |
| 3.1 | -4.09 |
| 3.15 | -4.13 |
| 3.2 | -4.16 |
| 3.25 | -4.18 |
| 3.3 | -4.20 |
| 3.35 | -4.22 |
| 3.4 | -4.22 |
| 3.45 | -4.46 |
Table 1. Limit example points.
The points on the graph correspond to the points in the table. You can see from both the graph and table that as \(x\) gets closer and closer to \(x= 3\), the function values get closer and closer to \(-4\) That means that
\[lim_{x \rightarrow 3} f(x)=-4\]
.
Notice that you don't actually care about the function value at \(x=3\) when finding the limit, because the definition says to look close to \(x=3\) but not at \(x=3\).
Finding Limits Algebraically
There are more examples of finding limits algebraically in a separate article. See Finding Limits of Specific Functions.
In fact, limits and continuity also go together.
If a function is continuous at a point, then the limit of the function exists and is equal to the function value at that point.
From the previous example, we had
\[f(x)=\dfrac{1}{4}(x+1)(x-1)(x-5)\]
and found the limit as \(x \rightarrow 3\). Since you know that all polynomials are continuous everywhere (see Continuity and see Theorems of Continuity for more details), you know the limit of the function exists and is equal to the function value. Since \(f(3)=-4\), that means
\[lim_{x \rightarrow 3} =-4\]
Look at the function
\[f(x)=\dfrac{x^2-2x-8}{x-4}\]
and find the limit as \(x \rightarrow 4\).
Answer:
The function is undefined at \(x=4\), so you can't just plug in the function value to find the limit. But you can factor the numerator to get
\[f(x)=\dfrac{x^2-2x-8}{x-4}=\dfrac{(x-4)(x+2)}{x-4}=x+2\]
as long as \(x \neq 4\). That means the graph of the function is actually the straight line \(y=x+2\) with a hole at the point \((4, 6)\). So \(lim_{x \rightarrow 4} f(x)=6\).
Finding the Derivative Using the Limit Definition
Using the definition of the derivative does involve limits. This is a big topic and it has a whole article on its own! See our article on Derivatives for lots more details on how to find the derivative using a limit.
Finding Limits - Key takeaways
- For any polynomial \(f(x), lim_{x \rightarrow \infty} f(x)=f(a)\).
- A table or graph can be used to find the limit of a function.
- Finding limits algebraically can involve factoring the numerator and denominator and seeing if anything cancels out. This is especially useful in cases where there is a hole in the graph.
- Properties of Limits can also be used to take the limit of functions.
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