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Applications of One-Sided Limits

Determine if a limit exists and, if so, its value.

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Applications of One-Sided Limits

Early in this section, we practiced finding one-sided limits. In this lesson we will be using that skill and applying it with the rule of limits that says a function must have the same limit from each side in order to have a single limit.

That process allows us to first determine if a function has as limit, and then find the limit if it exists, even if we cannot actually determine the limit directly.

Watch This

Embedded Video:

- James Sousa: Graphing Quadratics

Guidance

When we wish to find the limit of a function f(x) as it approaches a point a and we cannot evaluate f(x) at a because it is undefined at that point, we can compute the function's one-sided limits in order to find the desired limit. If its one-sided limits are the same, then the desired limit exists and is the value of the one-sided limits. If its one-sided limits are not the same, then the desired limit does not exist. This technique is used in the examples below.

Conditions For a Limit to Exist (The relationship between one-sided and two-sided limits)
In order for the limit L of a function to exist, both of the one-sided limits must exist at x0 and must have the samevalue. Mathematically,
if and only if and .
The One-Sided Limit
If f(x) approaches L as x approaches x0from the left and from the right, then we write
which reads: “the limit of f(x) as x approaches (or ) from the right (or left) is L.

Example A

Find the limit f(x) as x approaches 1. That is, find if

Solution

Remember that we are not concerned about finding the value of f(x) at x but rather near x. So, for x < 1 (limit from the left),

and for x > 1 (limit from the right),

Now since the limit exists and is the same on both sides, it follows that

Example B

Find .

Solution

From the figure below we see that decreases without bound as x approaches 2 from the left and increases without bound as x approaches 2 from the right.

This means that and . Since f(x) is unbounded (infinite) in either directions, the limit does not exist.

Example C

For an object in free fall, such as a stone falling off a cliff, the distance y(t) (in meters) that the object falls in t seconds is given by the kinematic equation y(t) = 4.9 t2. The object’s velocity after 2 seconds is given by .

What is the velocity of the object after 2 seconds?

Solution

The limit is 19.6 secs. The function can be plotted on a graphing tool, and at 1.999, the graph looks like this:

You can see the result of smaller values of t, by adjusting the t slider on the active graph here: https://www.desmos.com/drive/calculator/1ombivqkdl

Vocabulary

One-sided limits are limits of a function based on an approach from each direction individually.

Guided Practice

Questions

1) Find .
2) Find .
3) Find .
4) Find .
5) Find .

Solutions

1) If a and k are real numbers, then .
2) If a and k are real numbers, then .
3) The limit is 4, as shown in the image below. The red line approaches from values above x = 2, and the green line from below. The line is undefined where they meet. This can be examined in greater detail at: https://www.desmos.com/drive/calculator/oowpxjxeu2
4) The limit is
Interact with the graph here: https://www.desmos.com/drive/calculator/jqxhysqmwy
or make a table:
x
5
7
5.5
6.5
f(x)
1/11
1/13
2/23
2/25
5) The limit is or
interact with the graph here: https://www.desmos.com/drive/calculator/2ohonznchx

Practice

Based on the graph determine if a limit exists:

Determine if a limit exists:

  1. .
  2. Show that .

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