Limits and Continuity – Limits: Find Limits from Graphs

In the article “Introduction To Limits and Definition of Limits”, we were introduced to the idea of limits and how to find limits graphically and numerically. In this section, we will go more in depth on how to find the limits of functions from graphs by looking at a variety of scenarios.

There are three main discontinuities that you will encounter when graphically analyzing the limits of functions.

Removable Discontinuities

If there is a removable discontinuity (also known as a ‘hole’) in the curve of the graph at x = c, then the limit does exist on the graph of a function. Below is the example of this:

Notice that although the limit of F(x) as x approaches 1 is y = ⅓, the value of f(1) does not exist. The removable discontinuity means that there is no value at that specific point.

Infinite Discontinuity

If the given graph has a vertical asymptote at x = c and the curve from either the left or right side is increasing or decreasing without bound as x approaches a value c, then the limit does NOT exist. Below is an example of this.

As x is approaching zero from the left and right side, the curve increases without bound (+\infty), so the graph does not approach a particular value.

Jump Discontinuities

If the graph is approaching two different y values from two different directions as x approaches c, then the limit does not exist. There cannot be two different numbers. Below is an example of this:

The limit of F(x) from the left side as x approaches zero is -1, and from the right side 1. Since the limit of F(x) as x approaches zero from both the left and right side are different, the limit of F(x) as x approaches zero does NOT exist.

How To Analyze Limits Graphically

In order for a function to have a limit at x = c,

In other words, the limit of the function from both the left and right side must be the same.

Ask yourself this question: Is the limit of the function as F(x) approaches x = c the same from the left and right side?

  • If yes, then the function does have a limit as F(x) approaches c.
  • If no, then the function does not have a limit as F(x) approaches c.

Example 1: Suppose we have a function F(x) = x+2. What is the limit of the function as x approaches 3?

Solution:

Let’s graph the function.

We have to follow the path of the curve from both the left side and right side. If there is a convergence, then there is a limit.

The limit from both the left and right side are both the same: y = 5. Therefore, the limit of the function as x approaches 3 is y = 5. F(3) is also 5.

Example 2: Below is the graph of h(x):

Find the reasonable estimate for the limit of the h(x) at value of x = 3.

Solution:

In this graph, whenever we get closer to the value of x=3 from both the left and right side of the curve, the value of the function will also get closer to the 2, that’s why a reasonable estimate for the limit of h is 2. Notice, however, that this is a removable discontinuity (hole), so even though there is a limit at x = c, the value of the function at x = c does not exist.

Example 3: Below is a graph of g(x):

Find the reasonable estimate for the limit of the g(x) at value of x = 1.

Solution:

Let’s follow the path of the curve from both the left and right side as x approaches 1:


The limit seems to be somewhere between y = -2 and y = -1. Any value in between that range would be a good estimate (i.e. y = -1.5).

Example 4: You are given the graph having the function h(x)

Choose the correct answers from the given options.

Solution:

Let’s examine option A, which asks for the limit of F(x) as x approaches -2. We see that from both the left and right side, the function approaches a y value of 1, not 2. Therefore, option A is not correct.

Let’s examine option B, which asks for the limit of F(x) as x approaches 3 from the right side. Following the path of the curve from the right side, we see that the function approached a y value of 1. Therefore, option B is correct.

Let’s examine option C, which asks for the limit of F(x) as x approaches 2 from the left side. Following the path of the curve from the left side, the function approaches a y value of 1. Therefore, there is a limit, and option C is not correct.

Summary of Section

A discontinuity A region in the graph where F(x) experiences a distinct break or interruption.

There are 3 common types of discontinuities:

  1. Removable (hole): the limit of the function at x = c exists, but the value of the function at x = c doesn’t not exist.
  2. Jump: The function F(x) approaches two different y values from the left and right side.
  3. Infinite: a vertical asymptote exists at x = c and the graph either increases or decreases without bound as F(x) approaches x = c.

Determine if the limit of a function exists

The limit of F(x) as x approaches c from both the left and right side have to be the same:

References:
https://www.dummies.com/education/math/calculus/how-to-find-the-limit-of-a-function-graphically/
https://www.khanacademy.org/math/ap-calculus-ab/ab-limits-new/ab-1-3/a/approximating-limit-values-from-a-graph
https://sciencing.com/limit-exists-graph-of-function-4937923.html
https://www.khanacademy.org/math/ap-calculus-ab/ab-limits-new/ab-1-3/e/connecting-limits-and-graphical-behavior

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