Have you ever heard the saying “close only counts in horseshoes and hand grenades”? Well, it turns out, this isn't entirely true. Close, or nearly reaching a target, also counts in calculus – when dealing with limits, that is!
Basic Concept of a Limit in Mathematics
The basic concept of a limit in mathematics is essential to your understanding of calculus.
This concept has been around for thousands of years; early mathematicians used this concept to find better and better approximations of the area of a circle, for example.
The formal definition of a limit, however, has only been around since the 19th century. So, to begin your journey to understand limits, you should start with an intuitive definition.
Intuitive Definition of a Limit
To find an intuitive definition of a limit, you must first have a function (or several functions) about which you wish to know more details.
Take a look at the graphs of the following functions:
\[ f(x) = \frac{x^{2}-4}{x-2}, \; g(x) = \frac{|x-2|}{x-2}, \; \mbox{ and } \; h(x) = \frac{1}{(x-2)^{2}} \]
You want to pay attention to the behavior of these graphs at and approaching the value of \( x=2 \).
Pay attention to the graph where \( x = 2 \).
Pay attention to the graph where \( x = 2 \).
Pay attention to the graph where \( x = 2 \).
The graphs of these functions show their behavior at and around \( x=2 \). After observing them, can you see what they have in common?
They are all undefined when \( x=2 \)!
- But if that is all you say about them, you don't get very much information, do you? If you are given only this information, then, for all you know, all three of these functions could look identical. Based on their graphs, however, you know this isn't the case.
So, how can you express the behavior of these graphs more completely?
- With the use of limits, of course!
Now, take a closer look at how \( f(x) = \frac{x^2-4}{x-2} \) behaves near \( x = 2 \). Notice that as the values of \( x \) approach \( 2 \) from either side of \( 2 \), the values of \( f(x) \) approach \( 4 \).
To state this fact in mathematical terms, you would say: “the limit of \( f(x) \) as \( x \) approaches \( 2 \) is \( 4 \)”.
This statement is represented in mathematical notation as:
\[ \lim_{x \to 2} f(x) = 4. \]
From here, you can start to develop your intuitive definition of a limit – by thinking of the limit of a function at a number \( a \) as being the real number \( L \) that the functional values approach as its \( x \)-values approach \( a \), provided that the number \( L \) exists. More formally, this can be written as:
Let \( f(x) \) be a function that is defined at all values in an open interval containing \( a \) (possibly except \( a \)), and let \( L \) be a real number. If all values of \( f(x) \) approach the real number \( L \) as the values of \( x \) – except \( x = a \) – approach the number \( a \), then you can say that the limit of \( f(x) \) as \( x \) approaches \( a \) is \( L \).
Or, more simply:
As \( x \) gets closer and closer to \( a \), \( f(x) \) gets closer and closer and stays close to \( L \).
The idea of the limit is represented using mathematical notation as:
\[ \lim_{x \to a} f(x) = L \]
As you can see, just getting close to – or approaching – a point is how limits work! To develop and understand the key aspects of calculus, you first need to be comfortable with limits and the fact that approximations – or getting close to, or approaching, the desired value – are the basis of calculus. So, now you can change the saying from:
- “close only counts in horseshoes and hand grenades” to
- “close only counts in horseshoes, hand grenades, and calculus”!
Solving Limits
Before diving into algebraic methods, the next step to take intuitively is to develop a way for solving limits by estimating them. You can do this in one of two ways:
Solving a limit using a table of functional values
Solving a limit using a graph
Solving a Limit Using a Table of Functional Values
To solve a limit using a table of functional values, you can use this problem-solving strategy.
Strategy – Solving a Limit Using a Table of Functional Values
- If you want to solve the limit: \( \lim_{x \to a} f(x) \), you start by making a table of functional values.
- You should choose \( 2 \) sets of \( x \)-values – one set of values approaching \( a \) that are less than \( a \), and one set of values approaching \( a \) that are greater than \( a \). The table below gives an example of what your table could look like.
| Values Approaching \( a \) that are \( < a \) | | Values Approaching \( a \) that are \( > a \) |
|---|
| \( \bf{ x } \) | \( \bf{ f(x) } \) | | \( \bf{ x } \) | \( \bf{ f(x) } \) |
| \( a - 0.1 \) | \( f(a - 0.1) \) | | \( a + 0.1 \) | \( f(a + 0.1) \) |
| \( a - 0.01 \) | \( f(a - 0.01) \) | | \( a + 0.01 \) | \( f(a + 0.01) \) |
| \( a - 0.001 \) | \( f(a - 0.001) \) | | \( a + 0.001 \) | \( f(a + 0.001) \) |
| \( a - 0.0001 \) | \( f(a - 0.0001) \) | | \( a + 0.0001 \) | \( f(a + 0.0001) \) |
| Add more values if you need to. | | Add more values if you need to. |
- Next, look at the values in each of the columns labeled \( f(x) \).
- Determine if the values are approaching a single value as you move down each column.
- If both columns approach a common value, then you can say that\[ \lim_{x \to a} f(x) = L. \]
Solving a Limit Using a Graph
You can extend the problem-solving strategy above to solve a limit using a graph.
Strategy – Solving a Limit Using a Graph
- After following the above strategy, you can confirm your result by graphing the function.
- Using a graphing calculator (or other software), graph the function in question.
- Make sure the functional values of \( f(x) \) for the \( x \)-values near \( a \) are in the graphing window.
- Move along the graph of the function and check the \( y \)-values as their corresponding \( x \)-values approach \( a \).
- If the \( y \)-values approach \( L \) as the \( x \)-values approach \( a \) from both directions, then\[ \lim_{x \to a} f(x) = L. \]
Note that you may need to zoom in on the graph and repeat these steps several times.
For more details and examples, please refer to the articles on finding limits and finding limits using a graph or table.
Types of Limits
While the two techniques above are intuitive, they are inefficient and rely on too much guesswork to get the job done. But how can you progress past these methods?
Well, you will need to learn methods to solve, or evaluate, limits that are more algebraic in nature.
And how can you do that? First, you must know about two special limits; they provide the foundation of the algebraic methods to solve limits.
Ah, but what is so special about these two limits? These two limits are also known as basic limits, as they provide the basis for the limit laws. When you look at the graphs below, what do you notice?
No matter where along the line \( y = x \) the point \( (a, a) \) is, the limit as \(x\) approaches \(a\) is always \(a\).
No matter where along the line \( y = c \) the point \( (x, c) \) is, the limit as \(x\) approaches any real number \(a\) is always \(c\).
Based on these graphs, you can write out, algebraically, what the limit of these functions are. The algebraic interpretations of these are summarized in the theorem below.
Theorem: Basic Limits
Let \( a \) be a real number. Let \( c \) be a constant. Then:
\[ \begin{align}1. \; & \lim_{x \to a} x = a \\2. \; & \lim_{x \to a} c = c\end{align} \]
You can observe the following about these two limits:
- Notice that as \( x \) approaches \( a \), so does \( f(x) \).
- Consider the table:
| Values Approaching \( a \) that are \( < a \) | | Values Approaching \( a \) that are \( > a \) |
|---|
| \( \bf{ x } \) | \( \bf{ f(x) = c } \) | | \( \bf{ x } \) | \( \bf{ f(x) = c } \) |
| \( a - 0.1 \) | \( c \) | | \( a + 0.1 \) | \( c \) |
| \( a - 0.01 \) | \( c \) | | \( a + 0.01 \) | \( c \) |
| \( a - 0.001 \) | \( c \) | | \( a + 0.001 \) | \( c \) |
| \( a - 0.0001 \) | \( c \) | | \( a + 0.0001 \) | \( c \) |
- Notice that for all values of \( x \) – whether they are approaching \( a \) or not – the values of \( f(x) \) remain constant at \( c \).
- Therefore, \( \lim_{x \to a} c = c \)
Limit Rules
Building on these first two basic limit rules, the limit rules (also called limit laws) are listed below.
Theorem: Limit Laws
Let \( f(x) \) and \( g(x) \) be defined for all \( x \neq a \) over an open interval containing \( a \). Assume that \( L \) and \( M \) are real numbers, such that:
\[ \lim_{x \to a} f(x) = L \]
and\[ \lim_{x \to a} g(x) = M \]
Let \( c \) be a constant. Then the following are true:
Sum law for limits:
\[ \lim_{x \to a} (f(x) + g(x)) = \lim_{x \to a} f(x) + \lim_{x \to a} g(x) = L + M \]
Difference law for limits:
\[ \lim_{x \to a} (f(x) - g(x)) = \lim_{x \to a} f(x) - \lim_{x \to a} g(x) = L - M \]
Constant multiple law for limits:
\[ \lim_{x \to a} (c \cdot f(x)) = c \cdot \lim_{x \to a} f(x) = cL \]
Product law for limits:
\[ \lim_{x \to a} (f(x) \cdot g(x)) = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x) = L \cdot M \]
Quotient law for limits:
\[ \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)} = \frac{L}{M} \mbox{ where } M \neq 0\]
Power law for limits:
\[ \lim_{x \to a} (f(x))^{n} = \left( \lim_{x \to a} f(x) \right)^{n} = L^{n} \mbox{ for every positive integer } n \]
Root law for limits:
\[ \lim_{x \to a} \sqrt[n]{f(x)} = \sqrt[n]{\lim_{x \to a} f(x)} = \sqrt[n]{L} \mbox{ for all } L \mbox{ if } n \mbox{ is odd, and for } L \geq 0 \mbox{ if } n \mbox{ is even} \]
Keep in mind that there are other limit laws – the squeeze theorem and the intermediate value theorem. Please refer to those articles for more information.
The Existence of a Limit – When Does a Limit Not Exist?
When you work through the following example, remember that for the limit to exist, the functional values must approach a single real number value; otherwise the limit does not exist.
Evaluating a Limit that Does Not Exist (DNE) Due to Oscillations
Try to evaluate
\[ \lim_{x \to 0} sin \left( \frac{1}{x} \right) \]
using a table of functional values.
Solution:
- Create a table of values.
| \(\bf{x}\) | \(\bf{sin\left(\frac{1}{x}\right)}\) | | \(\bf{x}\) | \(\bf{sin\left(\frac{1}{x}\right)}\) |
| \(-0.1\) | \(0.54402\) | | \(0.1\) | \(-0.54402\) |
| \(-0.01\) | \(0.50636\) | | \(0.01\) | \(-0.50636\) |
| \(-0.001\) | \(-0.82688\) | | \(0.001\) | \(0.82688\) |
| \(-0.0001\) | \(0.30561\) | | \(0.0001\) | \(-0.30561\) |
| \(-0.00001\) | \(-0.03575\) | | \(0.00001\) | \(0.03575\) |
| \(-0.000001\) | \(0.34999\) | | \(0.000001\) | \(-0.34999\) |
- Carefully examine the table. What do you notice?
- The \( y \)-values aren't approaching any value. So, it seems like this limit doesn't exist. Before coming to this conclusion, though, you should take a systematic approach.
- Consider the following \( x \)-values for this function that approach \( 0 \):\[ \frac{2}{\pi}, \frac{2}{3\pi}, \frac{2}{5\pi}, \frac{2}{7\pi}, \frac{2}{9\pi}, \frac{2}{11\pi}, \cdots \]
- Their corresponding \( y \)-values are:\[ 1, -1, 1, -1, 1, -1, \cdots\]
- Based on the results, it is safe to conclude that the limit does not exist. The mathematical way to write this is:\[ \lim_{x \to 0} sin \left( \frac{1}{x} \right) \, DNE \]Where DNE stands for Does Not Exist.
- Of course, it is always a good idea to graph the function to confirm your result. The graph of \( f(x) = sin \left( \frac{1}{x} \right) \) shows that the function oscillates more and more wildly between \( -1 \) and \( 1 \) as \( x \) approaches \( 0 \).
The limit: \( \lim_{x \to 0} sin \left( \frac{1}{x} \right) \) does not exist because the function oscillates wildly as \( x \) approaches the limit of \( 0 \).
One-Sided Limits
There are times when saying that the limit of a function does not exist at a point does not provide enough information about that point. To see this, take another look at the second function from the beginning of this article.
\[ g(x) = \frac{|x-2|}{x-2} \]
As you choose values of \( x \) that are closer and closer to \( 2 \), \( g(x) \) does not approach a single value, but rather two values. Therefore, the limit does not exist, i.e.,
\[ \lim_{x \to 0} g(x) \, DNE. \]
While this statement is true, wouldn't you say that it doesn't quite give the full picture of the behavior of \( g(x) \) at \( x = 2 \)?
With one-sided limits, you can provide a more accurate description of the behavior of this function at \( x = 2 \).
For all values of \( x \) to the left of \( 2 \) – or the negative side of \( 2 \) – \( g(x) = -1 \).
So, you say that as \( x \) approaches \( 2 \) from the left, \( g(x) \) approaches \( -1 \). This is represented using mathematical notation as:
\[ \lim_{x \to 2^{-}} g(x) = -1 \]
For all values of \( x \) to the right of \( 2 \) – or the positive side of \( 2 \) – \( g(x) = 1 \).
So, you say that as \( x \) approaches \( 2 \) from the right, \( g(x) \) approaches \( 1 \). This is represented using mathematical notation as:
\[ \lim_{x \to 2^{+}} g(x) = 1 \]
Infinite Limits
Revisiting the third function from the beginning of this article, you will see there is a need to describe the behavior of functions that don't have finite limits.
\[ h(x) = \frac{1}{(x-2)^{2}} \]
From the graph of this function, you can see that as the values of \( x \) approach \( 2 \), the values of \( h(x) \) do not approach a value, but rather grow larger and larger, becoming infinite. This is represented using mathematical notation as:\[ \lim_{x \to 2^{+}} h(x) = +\infty \]
It is important to understand that when you say a limit is infinite, that does not mean the limit exists. It is simply a more descriptive way to say how the limit does not exist. \( \pm \infty \) is not a real number, so any infinite limit is not a limit that exists.
In general, limits at infinity are defined as:
Three Types of Infinite Limits
- Infinite limit from the left: Let \( f(x) \) be a function defined at all values in an open interval \( (b, a) \).
- If the values of \( f(x) \) increase without bound as the values of \( x \) (where \( x < a \)), approach the number \( a \), then the limit as \( x \) approaches \( a \) from the left is positive infinity. This is written as:\[ \lim_{x \to a^{-}} f(x) = +\infty. \]
- If the values of \( f(x) \) decrease without bound as the values of \( x \) (where \( x < a \)), approach the number \( a \), then the limit as \( x \) approaches \( a \) from the left is negative infinity. This is written as:\[ \lim_{x \to a^{-}} f(x) = -\infty. \]
- Infinite limit from the right: Let \( f(x) \) be a function defined at all values in an open interval \( (a, c) \).
- If the values of \( f(x) \) increase without bound as the values of \( x \) (where \( x > a \)), approach the number \( a \), then the limit as \( x \) approaches \( a \) from the right is positive infinity. This is written as:\[ \lim_{x \to a^{+}} f(x) = +\infty. \]
- If the values of \( f(x) \) decrease without bound as the values of \( x \) (where \( x > a \)), approach the number \( a \), then the limit as \( x \) approaches \( a \) from the right is negative infinity. This is written as:\[ \lim_{x \to a^{+}} f(x) = -\infty. \]
- Two-sided infinite limit: Let \( f(x) \) be defined for all \( x \neq a \) in an open interval containing \( a \).
- If the values of \( f(x) \) increase without bound as the values of \( x \) (where \( x \neq a \)), approach the number \( a \), then the limit as \( x \) approaches \( a \) is positive infinity. This is written as:\[ \lim_{x \to a} f(x) = +\infty. \]
- If the values of \( f(x) \) decrease without bound as the values of \( x \) (where \( x \neq a \)), approach the number \( a \), then the limit as \( x \) approaches \( a \) is negative infinity. This is written as:\[ \lim_{x \to a} f(x) = -\infty. \]
Limits Examples
Use the limit laws to solve:
\[ \lim_{x \to -3} (4x+2) \]
Solution:
To solve this limit, apply the limit laws one at a time. Keep in mind that – at each step – you need to check that the limit exists before you apply the law. The new limit must exist for the law to be applied.
- Apply the sum law.\[ \lim_{x \to -3} (4x+2) = \lim_{x \to -3} 4x + \lim_{x \to -3} 2 \]
- Apply the constant multiple law.\[ \lim_{x \to -3} (4x+2) = 4 \cdot \lim_{x \to -3} x + \lim_{x \to -3} 2 \]
- Apply the basic limit.\[ \lim_{x \to -3} (4x+2) = 4 \cdot (-3) + 2\]
- Simplify.\[ \lim_{x \to -3} (4x+2) = -10\]
Limits – Key takeaways
- Limits are all about determining how a function behaves as it approaches a specific point or value.
- The mathematical notation for a limit is:\[ \lim_{x \to a} f(x) = L \]
- Intuitively, limits can be evaluated using a table of functional values, or the graph of the function.
- There are several limit laws that make evaluating limits much easier:
- Two Important Limits\[ \begin{align}1. \; & \lim_{x \to a} x = a \\2. \; & \lim_{x \to a} c = c\end{align} \]
- Sum law for limits:\[ \lim_{x \to a} (f(x) + g(x)) = \lim_{x \to a} f(x) + \lim_{x \to a} g(x) = L + M \]
- Difference law for limits:\[ \lim_{x \to a} (f(x) - g(x)) = \lim_{x \to a} f(x) - \lim_{x \to a} g(x) = L - M \]
- Constant multiple law for limits:\[ \lim_{x \to a} (c \cdot f(x)) = c \cdot \lim_{x \to a} f(x) = cL \]
- Product law for limits:\[ \lim_{x \to a} (f(x) \cdot g(x)) = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x) = L \cdot M \]
- Quotient law for limits:\[ \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)} = \frac{L}{M} \mbox{ where } M \neq 0\]
- Power law for limits:\[ \lim_{x \to a} (f(x))^{n} = \left( \lim_{x \to a} f(x) \right)^{n} = L^{n} \mbox{ for every positive integer } n \]
- Root law for limits:\[ \lim_{x \to a} \sqrt[n]{f(x)} = \sqrt[n]{\lim_{x \to a} f(x)} = \sqrt[n]{L} \mbox{ for all } L \mbox{ if } n \mbox{ is odd, and for } L \geq 0 \mbox{ if } n \mbox{ is even} \]
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