The Limit of A Function
What are Limits?
In calculus, the limit of a function is an important concept which stands as the foundation for more complex topics. It is a fundamental idea which provides a deeper understanding of how functions behave. Specifically, the concept of limits describes what happens to a function as it gets closer and closer to a particular value. The idea of limits is illustrated by the image below.
Note:
Limits are concerned with what happens to the function as you approach a point, not at the point.
Let us observe the behaviour of the function f(x) around the point x = 1. As shown, as we approach x = 1 from either the left or right side of the function, the value of the function (y value) approaches 2. So, we can say that the limit of f(x) as x approaches 1 is 2. This is the idea of the limit in simple terms and this concept allows us to analyze the behaviour of functions as they get closer and closer to a certain point.
Limit Notation
When discussing and evaluating limits, there is a certain notation that must be used. See the notation below. This reads as “the limit of f(x) as x approaches a, equals L”. Essentially, the values of f(x) get closer and closer to L as x gets closer and closer to a.
Notation:
[latex]\Large{\lim_{x\to a}f(x) = L}[/latex]
Now, let’s take a look at another function, suppose [latex]f(x) = x^2+3 [/latex]. What is the behaviour of this function as x approaches 2? The table below shows the values of the function f(x) for different values of x close to 2.
As you can see, as x is near 2 on either side, the value of f(x) goes closer and closer to 7. We can then say, “the limit of the function [latex]f(x) = x^2+3[/latex] as x approaches 2 is equal to 7.
Expressing this in limit notation, we can write:
[latex]\Large{\lim_{x\to 2}(x^2+3) = 7}[/latex]
Example 1
Find the following limit:
[latex]\LARGE{\lim_{x\rightarrow 2}(2x)}[/latex]
Step 1:
Let us first graph the function, as shown below.
Step 2:
Now, notice that as the x value approaches x = 2, the value of the function approaches y = 4. So, we can say that the limit of [latex]f(x) = 2x[/latex] as x approaches 2, is 4.
Step 3:
Expressing this in limit notation, we have:
[latex]\large{\lim_{x \to 2}(2x) = 2(2) = 4}[/latex]
Step 1:
Let us first graph the function, as shown below.
Step 2:
Now, notice that as the x value approach x = 2, the value of the function approaches y= 4. So, we can say that the limit of [latex]f(x) = 2x[/latex] as x approaches 2, is 4/
Step 3:
Expressing this in limit notation, we have:
[latex]\large{\lim_{x\to 2}(2x) = 2(2) = 4}[/latex]
Need more explanation? Chat with a tutor now!
QUESTIONS?
If you have any questions about our services or have any feedback, do not hesitate to get in touch with us!