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AP®︎/College Calculus AB

Unit 1: Lesson 2

Defining limits and using limit notation

Limits intro

AP.CALC:
LIM‑1 (EU)
,
LIM‑1.A (LO)
,
LIM‑1.A.1 (EK)
,
LIM‑1.B (LO)
,
LIM‑1.B.1 (EK)
Limits describe how a function behaves near a point, instead of at that point. This simple yet powerful idea is the basis of all of calculus.
To understand what limits are, let's look at an example. We start with the function f, left parenthesis, x, right parenthesis, equals, x, plus, 2.
Function f is graphed. The x-axis goes from 0 to 9. The graph consists of a line starting at (0, 2) and moving upward through (2,4) and (4, 6), and ending at (7,9).web+graphie://cdn.kastatic.org/ka-perseus-graphie/507e8f38d9db338d657f07b535ba2ed4a8a9d206
The limit of f at x, equals, 3 is the value f approaches as we get closer and closer to x, equals, 3. Graphically, this is the y-value we approach when we look at the graph of f and get closer and closer to the point on the graph where x, equals, 3.
For example, if we start at the point left parenthesis, 1, comma, 3, right parenthesis and move on the graph until we get really close to x, equals, 3, then our y-value (i.e. the function's value) gets really close to 5.
The graph of function f is animated. A point moves upward on the line from (1, 3) to (2.99, 4.99).
Created with Geogebra.
Similarly, if we start at left parenthesis, 5, comma, 7, right parenthesis and move to the left until we get really close to x, equals, 3, the y-value again will be really close to 5.
The graph of function f is animated. A point moves downward on the line from (5, 7) to (3.01, 5.01).
Created with Geogebra.
For these reasons we say that the limit of f at x, equals, 3 is 5.
The graph of function f has arrows pointing along the line, pointing up to the right and down to the left, respectively, pointing to point (3, 5).
You might be asking yourselves what's the difference between the limit of f at x, equals, 3 and the value of f at x, equals, 3, i.e. f, left parenthesis, 3, right parenthesis.
So yes, the limit of f, left parenthesis, x, right parenthesis, equals, x, plus, 2 at x, equals, 3 is equal to f, left parenthesis, 3, right parenthesis, but this isn't always the case. To understand this, let's look at function g. This function is the same as f in every way except that it's undefined at x, equals, 3.
Function g is graphed. The x-axis goes from 0 to 9. The graph consists of a line starting at (0, 2), moving upward through (2, 4) and an open circle at (3, 5), and ending at (7, 9).
Just like f, the limit of g at x, equals, 3 is 5. That's because we can still get very very close to x, equals, 3 and the function's values will get very very close to 5.
The graph of function g has arrows pointing along the line, pointing up to the right and down to the left, respectively, pointing to the open circle at (3, 5).
So the limit of g at x, equals, 3 is equal to 5, but the value of g at x, equals, 3 is undefined! They are not the same!
That's the beauty of limits: they don't depend on the actual value of the function at the limit. They describe how the function behaves when it gets close to the limit.
Problem 1
This is the graph of h.
What is a reasonable estimate for the limit of h at x, equals, 3?

We also have a special notation to talk about limits. This is how we would write the limit of f as x approaches 3:
\begin{aligned} \scriptsize\text{"The limit of..."}&\qquad\scriptsize\text{"...the function }f\text{..."} \\ \searrow\qquad&\qquad\swarrow \\ \LARGE\displaystyle\lim_{x\to 3}&\LARGE f(x) \\ \nearrow\qquad \\ \scriptsize\text{"...as }x\text{ approaches }3\text{."} \end{aligned}
The symbol limit means we're taking a limit of something.
The expression to the right of limit is the expression we're taking the limit of. In our case, that's the function f.
The expression x, \to, 3 that comes below limit means that we take the limit of f as values of x approach 3.
Problem 2
This is the graph of f.
What is a reasonable estimate for limit, start subscript, x, \to, 6, end subscript, f, left parenthesis, x, right parenthesis ?

Problem 3
Which expression represents the limit of x, squared as x approaches 5?

In limits, we want to get infinitely close.

What do we mean when we say "infinitely close"? Let's take a look at the values of f, left parenthesis, x, right parenthesis, equals, x, plus, 2 as the x-values get very close to 3. (Remember: since we're dealing with limits we don't care about f, left parenthesis, 3, right parenthesis itself.)
xf, left parenthesis, x, right parenthesis
2, point, 94, point, 9
2, point, 994, point, 99
start color gray, start underbrace, start color black, 2, point, 999, end color black, end underbrace, start subscript, start text, c, l, o, s, e, space, t, o, space, end text, 3, end subscript, end color graystart color gray, start underbrace, start color black, 4, point, 999, end color black, end underbrace, start subscript, start text, c, l, o, s, e, space, t, o, space, end text, 5, end subscript, end color gray
We can see how, when the x-values are smaller than 3 but become closer and closer to it, the values of f become closer and closer to 5.
xf, left parenthesis, x, right parenthesis
3, point, 15, point, 1
3, point, 015, point, 01
start color gray, start underbrace, start color black, 3, point, 001, end color black, end underbrace, start subscript, start text, c, l, o, s, e, space, t, o, space, end text, 3, end subscript, end color graystart color gray, start underbrace, start color black, 5, point, 001, end color black, end underbrace, start subscript, start text, c, l, o, s, e, space, t, o, space, end text, 5, end subscript, end color gray
We can also see how, when the x-values are larger than 3 but become closer and closer to it, the values of f become closer and closer to 5.
Notice that the closest we got to 5 was with f, left parenthesis, 2, point, 999, right parenthesis, equals, 4, point, 999 and f, left parenthesis, 3, point, 001, right parenthesis, equals, 5, point, 001, which are 0, point, 001 units away from 5.
We can get closer than that if we want. For example, suppose we wanted to be 0, point, 00001 units from 5, then we would pick x, equals, 3, point, 00001 and then f, left parenthesis, 3, point, 00001, right parenthesis, equals, 5, point, 00001.
This is endless. We can always get closer to 5. But that's exactly what "infinitely close" is all about! Since being "infinitely close" isn't possible in reality, what we mean by limit, start subscript, x, \to, 3, end subscript, f, left parenthesis, x, right parenthesis, equals, 5 is that no matter how close we want to get to 5, there's an x-value very close to 3 that will get us there.
If you find this hard to grasp, maybe this will help: how do we know there are infinite different integers? It's not like we've counted them all and got to infinity. We know they are infinite because for any integer there's another integer that's even larger than that. There's always another one, and another one.
In limits, we don't want to get infinitely big, but infinitely close. When we say limit, start subscript, x, \to, 3, end subscript, f, left parenthesis, x, right parenthesis, equals, 5, we mean we can always get closer and closer to 5.
Problem 4
xg, left parenthesis, x, right parenthesis
minus, 7, point, 16, point, 32
minus, 7, point, 016, point, 1
minus, 7, point, 0016, point, 03
minus, 6, point, 9996, point, 03
minus, 6, point, 996, point, 1
minus, 6, point, 96, point, 32
What is a reasonable estimate for limit, start subscript, x, \to, minus, 7, end subscript, g, left parenthesis, x, right parenthesis?

Another example: $\displaystyle\lim_{x\to 2}x^2$limit, start subscript, x, \to, 2, end subscript, x, squared

Let's analyze limit, start subscript, x, \to, 2, end subscript, x, squared, which is the limit of the expression x, squared when x approaches 2.
Function y = x squared is graphed. The x-axis goes from negative 4 to 6. The graph consists of a curve. The curve is a parabola, starting at (negative 3, 9), moving downward through (negative 1, 1) to (0, 0), moving upward through (1, 1), and ending at (3, 9).
We can see how, when we approach the point where x, equals, 2 on the graph, the y-values are getting closer and closer to 4.
The graph of y = x squared is animated with a point moving up the curve from (1.5, 2.25) to (1.99, 3.96) and then moving down the curve from (2.5, 6.25) to (2.01, 4.04).
Created with Geogebra.
We can also look at a table of values:
xx, squared
1, point, 93, point, 61
1, point, 993, point, 9601
start color gray, start underbrace, start color black, 1, point, 999, end color black, end underbrace, start subscript, start text, c, l, o, s, e, space, t, o, space, end text, 2, end subscript, end color graystart color gray, start underbrace, start color black, 3, point, 996001, end color black, end underbrace, start subscript, start text, c, l, o, s, e, space, t, o, space, end text, 4, end subscript, end color gray
xx, squared
2, point, 14, point, 41
2, point, 014, point, 0401
start color gray, start underbrace, start color black, 2, point, 001, end color black, end underbrace, start subscript, start text, c, l, o, s, e, space, t, o, space, end text, 2, end subscript, end color graystart color gray, start underbrace, start color black, 4, point, 004001, end color black, end underbrace, start subscript, start text, c, l, o, s, e, space, t, o, space, end text, 4, end subscript, end color gray
We can also see how we can get as close as we want to 4. Suppose we want to be less than 0, point, 001 units from 4. Which x-value close to x, equals, 2 can we choose?
Let's try x, equals, 2, point, 001:
2, point, 001, squared, equals, 4, point, 004001
That's more than 0, point, 001 units away from 4. Alright, so let's try x, equals, 2, point, 0001:
2, point, 0001, squared, equals, 4, point, 00040001
That's close enough! By trying x-values that are closer and closer to x, equals, 2, we can get even closer to 4.
In conclusion, limit, start subscript, x, \to, 2, end subscript, x, squared, equals, 4.

A limit must be the same from both sides.

Coming back to f, left parenthesis, x, right parenthesis, equals, x, plus, 2 and limit, start subscript, x, \to, 3, end subscript, f, left parenthesis, x, right parenthesis, we can see how 5 is approached whether the x-values increase towards 3 (this is called "approaching from the left") or whether they decrease towards 3 (this is called "approaching from the right").
Function f is graphed. The x-axis goes from 0 to 9. The graph is a line starting at (0, 2) and moving upward through (2,4) and (4, 6). An arrow pointing up the line to (3, 5) represents approach from the left. An arrow pointing down the line to (3, 5) represents approach from the right.
Now take, for example, function h. The y-value we approach as the x-values approach x, equals, 3 depends on whether we do this from the left or from the right.
Function h is graphed. The x-axis goes from 0 to 9. The graph consists of 2 lines. The first line starts at (0, 1), moves upward, and ends at an open circle at (3, 4). The second line starts at a closed circle at (3, 6), moves upward, and ends at (6, 9).
When we approach x, equals, 3 from the left, the function approaches 4. When we approach x, equals, 3 from the right, the function approaches 6.
The graph of function h has an arrow, representing approach from the left, pointing up to the right along the first line to the open circle at (3, 4). Another arrow, representing approach from the right, points down to the left along the second line to the closed circle at (3, 6).
When a limit doesn't approach the same value from both sides, we say that the limit doesn't exist.
Problem 5
This is the graph of function g.
Which of the limits exists?

Want to join the conversation?

• In problem 5 why can one of the answers be x→6? but not x→3?
• I assume you are talking about the last example. As you approach x=6 from the left you move closer to 3; AND as you approach x=6 from the right, you also moce closer to 3.

As you move closer to x=3 form the left you move closer to 3, BUT when you move closer to x=3 from the right, you move closer to 6.

They must be moving to the same value of y from both sides if you are not going to specify the side in th limit notation.
• When we take a limit, we approach a specific x value from both sides. But there is an infinite number of numbers between any two numbers. So when we are approaching x, can we ever really get there? And if we can not, then does the calculus give us an answer with really really really really really small flaw that does not create problem in any calculation? Or is it 100% flawless?
• This is the issue that a lot of people had in the development of the calculus. The solution was quite clever, the idea is that you can get "arbitrarily close" that is given any epsilon positive there's a delta such that |x-a| < delta implies |f(x) - f(a)| < epsilon. If this is true FOR EVERY epsilon, then we say the limit exists and the function is continuous. Is it 100% flawless? Not really, but the thing is it's useful. Using these ideas we were able to build up a foundation for calculus which eventually lead to the physics that put man on the moon. The point is it gets results.
• For the last question, how is there a limit for x-->6? There is a point there. I thought there could only be limits if there were open dots.
• limits can exist for any point whether defined or undefined. it does not exist only when both the sides do not approach the same value for a point.
• Can a limit ever be "undefined" or is it strictly described as "none existent"? I am sure it comes down to the definitions of those two phrases/words but I want to hear your thoughts.
• A definition is just a description of our mathematical object. We can make up any definition we like.

We say an object is 'well-defined' (as opposed to 'undefined') when our definition actually pinpoints an object or class of objects. 'Prime numbers which are perfect squares' is a well-defined concept, because both prime numbers and perfect squares are well-defined. As it happens, no such numbers exist.

So 'the limit of sin(x) as x→∞' is a well-defined concept; it's the real number that satisfies the ε-δ definition of that limit. It's just that no such real number exists, so we say the limit doesn't exist.
• the plot has open dots and closed dots. what does that symbolize in limits?
• Open dots means it doesn't include that point and closed dots mean it does include that point. For example, a open dot at 6 means it can't be 6 but it can be 5.999999 or 6.000001, just not 6. A closed circle for the point 6 would mean it includes 6.
• In the last question, how does
x→7
​lim
​​ g(x)
exist? It has two locations right?
• @Rachel the closed cricle is just confusing you, the close circle means when x = 7 ,y-value is equal to 2. What we are doing is limits that is when x is apporaching 7, y is appraoching 4. it is appraoching not there, Even if you take close cirlce for a instance you will reach that close circle( 2 y-value) for either of sides, so yes limit exists. Hope that helps
• what is the difference between APcalculus AB and APcalculus BC?
• More specific course content is given on the College Board website.

Essentially, AB is equivalent to Calc 1, while BC is equivalent to Calc 1 and 2. AB covers limits, derivatives, and integrals. BC covers everything that AB does, in addition to derivatives of vector-valued functions, polar functions, parametric functions, planar motion, Euler's Method, improper integrals, integration by parts, arc length, polar areas, the logistic model, and (a whole unit on) series. Hope that I helped.
• what is the different between un defined and Does not exit?
• When limit of f(x) as x approaches ..say..’a’ has 2 different values when approaching x=a from the left and right, doesn't that basically mean for a specific input(x) there are 2 different outputs(y)

So, f(x) is not really a function
I hope I’ve not got my basics missed up(if so, do clarify),
Then wouldn’t a limit be defined for all real functions ?