Limits and Continuity – Introduction to Limits and Definitions of Limits

Suppose that Adam and Jeorge decided to meet at some outside park. Adam takes the road going east and Jeorge takes the road going west. Even though each person is using a different route to reach the same destination, ultimately, they both converge at the same location.

Finding the limits of functions is based on the same idea. Following the path of a curve from both the left and right side, the limit of a function is the y-value that the function approaches (but is not equal to) at a certain x-value. Examine the diagram below:

At x = c, there is a limit L, but F(c) does not exist (there is a removable discontinuity, or ‘hole’ at x = c, which we will determine how to obtain in a later section).

Definition (Informal): The Limit of a Function

Let F(x) be a function defined on the interval that contains x = c, except possibly at the x = c, then we can say that:

Important things to consider:

  • Sufficiently closed would consist of those values which are less than and greater than to c.
  • Some of the functions may or may not have the limits at specific values of a (see figure 1)
  • Notice that the value of the limit is not affected by F(c).

In other words, the limit is the y-value that the function ‘approaches’ as it gets closer to an x-value. The limit of a function describes the behavior of the function when the variable is near, but does not equal, a specified number. If the values of F(x) get closer and closer, as close as we want, to one letter as we take values of x very close to (but not equal to) a letter c, then the limit of that function is L.

Generally, there are three ways to analyze the limit of a function:

  1. Numerical approach (T – table)
  2. Graphical approach: Analyze the graph
  3. Algebraic approach: Use algebra or calculus

In this section we will focus on the graphical and numerical approach (the algebraic approach will be discussed in another section.

Let’s do an example of solving limits graphically.

Example 1: Use the graph of y = F(x) to find the following limits and function values:

Solution:

Let’s look at x = -1.

If we follow the path of the curve from both the left and right side, it approaches a y-value of y = 2. Therefore

To find the value of f(-1), we need to look at both the path of the curve and the solid dot. Following the curve, there is a removable discontinuity, or ‘hole’ for the point (-1,2), which means there is no value at that point. Therefore, the value of f(-1) cannot be 2. However, we do have a value at x = -1, which is y = 1. Therefore, f(-1)= 1.

Let’s look at x = -4.

If we follow the path of the curve from the left side, the function approaches y = 3. When we follow the path of the curve from the right side, the function approaches y = 4. This is because the limit from both the left and right side are NOT the same,  (does not exist). To find the value of f(-4), we need to follow the path from both the left and right side again. From the right side, the limit of F(x) is 4. However, there is another ‘hole’ there, so there is no value. From the left side, the limit is 3. Since there is a point at the end of the path, the y value at that point would be the value of f(-4), which is 3.

Let’s look at x = 4.

If we follow the path of the curve from the left side, the function approaches y = -3. When we follow the path of the curve from the right side, the function approaches y = -2. Because the limit from both the left and right side are NOT the same, DNE (does not exist). To find the value of f(4), we need to follow the path from both the left and right side again. If we follow the path from the left side, the limit of F(x) is -3. However, there is another ‘hole’ there, so there is no value. Therefore, the value of f(4) cannot be -3. From the right side, the limit of F(x) is -2 because there is a point at the end of the path (4, -2), f(4)= 2.

Let’s do an example of solving limits numerically:

Example 2: Estimate the limit of the following function by using a data table. Graph the function on a calculator or computer to verify your results. Record to at least 4 decimal places.

Solution:

Construct the data table:

x 3.9 3.99 3.999 4 4.001 4.01 4.1
F(x)

Notice that I selected x values that were greater than and less than 4 to precisely estimate the limit.

Let’s find the values of f(3.9), f(3.99), and f(3.999) as well as f(4.001), f(4.01), and f(4.1):

  • f(3.9) = 0.2041
  • f(3.99) = 0.2004
  • f(3.999) = 0.2
  • f(4.001) = 0.2
  • f(4.01) = 0.1996
  • f(4.1) = 0.1961

Let’s now fill out the data table:

x 3.9 3.99 3.999 4 4.001 4.01 4.1
F(x) 0.2041 0.2004 0.2 ?? 0.2 0.1996 0.1961

Looking at the data table from both the left and right sides, as the function approaches x = 4, the function F(x) approaches y = 0.2.

x 3.9 3.99 3.999 4 4.001 4.01 4.1
F(x) 0.2041 0.2004 0.2 0.2 0.2 0.1996 0.1961

One-Sided Limits

From the example problems, we have learned that in order for a function F(x) to have a limit at x = c, the limit from both the left and right side must be the same. If the limit of the function from both sides of the curve are NOT the same, then there is no limit of the function at x = c.

Theorem: One – Sided Limits

As the x arrives the c, the limit of the function F(x) is L if the limit from the left side present and limit from the right side present are L that is if:

Example 3: Use the graph of y = F(x) to find the following limits and function values:

Let’s look at x = 1.

Following the path of the curve from the left side, the limit of F(x) as x approaches 1 is y = 3. Following the path of the curve from the right side, the limit of F(x) as x approaches 1 is also y = 3. Therefore, the limit of F(x) as x approaches 1 is y = 3.

Following the path of the curve from the left side, the limit of F(x) as x approaches -5 is y = 2. Following the path of the curve from the right side, the limit of F(x) as x approaches -5 is y = 3. Because the limit from both the left and right side are NOT the same, the limit of F(x) as x approaches -5 does not exist (DNE).

Therefore,

Let’s look at x = 5.

Following the path of the curve from the left side, the limit of F(x) as x approaches -5 is y = 0. Following the path of the curve from the right side, the limit of F(x) as x approaches -5 is y = 1. Because the limit from both the left and right side are NOT the same, the limit of F(x) as x approaches -5 does not exist (DNE). Therefore,

References:
https://www.khanacademy.org/math/ap-calculus-ab/ab-limits-new/ab-1-2/e/limits-intro
http://tutorial.math.lamar.edu/Classes/CalcI/DefnOfLimit.aspx
https://math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/SandS/lHopital/define_limit.html
https://www.toppr.com/guides/maths/limits-and-derivatives/limits/
http://www.la-citadelle.com/courses/calculus/lim08.pdf
http://scidiv.bellevuecollege.edu/dh/Calculus_all/CC_1_1_limit

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