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(x)=x+2.
The limit f at x=3 is the value f approaches as we get closer and closer to x=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=3.
For example, if we start at the point (1,3) and move on the graph until we get really close to x=3, then our y-value (i.e. the function's value) gets really close to 5.
Similarly, if we start at (5,7) and move to the left until we get really close to x=3, the y-value again will be really close to 5.
For these reasons we say that the limit of f at x=3 is 5.
You might be asking yourselves what's the difference between the limit of f at x=3 and the value of f at x=3, i.e. f(3).
So yes, the limit of f(x)=x+2 at x=3 is equal to f(3), but this isn't always the case. To understand this, let's look at the function g. This function is the same as f in every way except that it's undefined at x=3.
What kind of function is g anyway?
Well, you can look at g as if it's a piecewise function:
g(x)={x+2undefinedif x=3if x=3
You can also define g to be the rational function x−3x2−x−6. This function isn't defined at x=3, because this results in a division by zero. On the other hand, it simplifies into f:
g(x)=x−3x2−x−6
=x−3(x+2)(x−3)
=x+2,for x=3
Just like f, the limit of g at x=3 is 5. That's because we can still get very very close to x=3 and the function's values will get very very close to 5.
So the limit of g at x=3 is equal to 5, but the value of g at x=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=3?
Choose 1:
2
3
4
The limit doesn't exist
Step by step solution
As we get closer to x=3, the function values get closer to 2. Therefore, a reasonable estimate for the limit of h at x=3 is 2.
We also have a special notation to talk about limits. This is how we would write the limit of f as x approaches 3:
The symbol lim means we're taking a limit of something.
The expression to the right of lim is the expression we're taking the limit of. In our case, that's the function of f.
The expression x→3 that comes below lim 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 the limit of x→6limf(x)?
Choose 1:
−5
−3
6
The limit doesn't exist
Step by step solution
As we get closer to x=6, the function values get closer to −3. Therefore, a reasonable estimate for x→6limf(x) is −3.
PROBLEM 3
Which expression represents the limit of x2as x approaches 5?
Choose 1:
lim52
x2→5lim
x→5limx2
x→25limx
Step by step solution
In limits, we want to get infinitely close.
What do we mean when we say "infinitely close"? Let's take a look at the value of f(x)=x+2 as the x-values get very close to 3. (Remember: since we're dealing with limits we don't care don't care about f(3) itself.)
x
f(x)
2.9
4.9
2.99
4.99
close to 32.999
close to 54.999
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.
x
f(x)
3.1
5.1
3.01
5.01
close to 33.001
close to 55.001
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(2.999)=4.999 and f(3.001)=5.001, which are 0.001 units away from 5.
We can get closer than that if we want. For example, suppose we wanted to be 0.00001 units from 5, then we would pick x=3.00001 and then f(3.0001)=5.0001.
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 x→3limf(x)=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 x→3limf(x)=5, we mean we can always get closer and closer to 5.
PROBLEM 4
x
g(x)
−7.1
6.32
−7.01
6.1
−7.001
6.03
−6.999
6.03
−6.99
6.1
−6.9
6.32
What is a reasonable estimate for the limit ofx→−7limg(x)?
Choose 1:
−7
6
6.15
6.33
The limit doesn't exist
Step by step solution
As the x-values increase towardsx=−7, the g(x)-values seem to approach 6.
x
g(x)
−7.1
6.32
−7.01
6.1
close to -7−7.001
close to 66.03
As the x-values decrease towardsx=−7, the g(x)-values seem to approach 6.
x
g(x)
−6.9
6.32
−6.99
6.1
close to -7−6.999
close to 66.03
So a reasonable estimate for x→−7limg(x) is 6.
Another example: x→2limx2
Let's analyze x→2limx2, which is the limit of the expression x2 when x approaches 2.
We can see how, when we approach the point where x=2 on the graph, the y-values are getting closer and closer to 4.
We can also look at a table of values:
x
x2
1.9
3.61
1.99
3.9601
close to 21.999
close to 43.996001
x
x2
2.1
4.41
2.01
4.0401
close to 22.001
close to 44.004001
We can also see how we can get as close as we want to 4. Suppose we want to be less than 0.001 units from 4. Which x-value close to x=2 can we choose?
Let's try x=2.001:
2.0012=4.004001
That's more than 0.001 units away from 4. Alright, so let's try x=2.0001:
2.00012=4.0004001
That's close enough! By trying x-values that are closer and closer to x=2, we can get even closer to 4.
In conclusion, x→2limx2=5.
A limit must be the same from both sides.
Coming back to f(x)=x+2 and x→3limf(x), 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").
Now take, for example, function h. The y-value we approach as the x-values approach x=3 depends on whether we do this from the left or from the right.
When we approach x=3 from the left, the function approaches 4. When we approach x=3 from the right, the function approaches 6.
When a limit doesn't approach the same value from both sides, we say the limit doesn't exist.
PROBLEM 5
This is the graph of function g.
Which of the limits exists?
Choose all that apply:
x→3limg(x)
x→5limg(x)
x→6limg(x)
x→7limg(x)
Step by step solution
The y-value we approach as we approach x=3 is different for each side.
Therefore, x→3limg(x) doesn't exist.
In contrast to x=3, the limit of g at the other x-values does exist, because we approach the same y-value from both sides. For example, when we approach x=6 from either side, the y-values approach 3.