Algebraic Properties of Limits

In this section, we will discuss some algebraic properties of limits that can help us evaluate limits of more complex functions.

No Proofs

Note that we will not be providing formal proofs for these properties. The proofs are typically more involved and require a deeper understanding of the concepts involved.

Generally, these properties can be proven using the formal definition of limits, which falls into the realm of real analysis.

Limit of a Sum

The limit of a sum of two functions is the sum of their limits;

This property can be extended to the sum of multiple functions:

Limit of Products

The limit of a product of two functions is the product of their limits:

Similarly, the limit of the product of a constant and a function is:

Limit of Powers

The limit of a power of a function is the power of the limit of the function:

Limit of Composite Functions

This is a bit more complex. Let's say we have and , and we want to find the limit of as approaches .

Limit with Conditions Met

Just like with the other properties, we can break this down into the limits of the individual functions:

However, this relies on two conditions:

1. exists. Call this limit . Otherwise, the inner limit on the right-hand side will not exist.
2. is continuous at .

Limit with Conditions Not Met

If either of these conditions is not met, we cannot use this property directly. However, just because this property doesn't work doesn't mean the limit doesn't exist.

In such cases, we may need to use other methods to evaluate the limit. Usually, we would need to consider the limit from both sides and see if they agree. One example problem below will illustrate this.

Example Problem: Graphically Evaluating Composite Limits

Two functions, and , are unknown. We are given the graphs of them:

Graph of

Graph of

Evaluate the following limit:

To evaluate this limit, we can use the properties of limits we have discussed.

First, ignore the term for now. Notice that there's a constant term multiplied by , and a added to the result. Through the properties of limits, we can extract these constants:

Next, since we have a composite function , we can check if the conditions for the limit of composite functions are met:

1. exists.

   From the graph of , we can see that as approaches , approaches . Even though there's a hole in the graph at , the limit still exists; from a graphical perspective, the curve approaches from both sides.

2. is continuous at .

   From the graph of , we can see that . This means that is continuous at .

Therefore, we can evaluate the limit of the composite function :

Finally, substitute this result back into the original limit:

Example Problem: Evaluating a Limit with a Discontinuous Function

Using the same functions and as in the previous example, evaluate the following limit:

In this case, we have the same functions and , but we are evaluating a different limit. Recall the conditions for the limit of composite functions:

1. exists.

   From the graph of , we can see that as approaches , approaches .

2. is continuous at .

   From the graph of , we can see that is not continuous at (there's a jump discontinuity).

Therefore, we cannot directly use the property of composite functions to evaluate this limit. If there's a discontinuity in the function, we need to consider the limit from both sides.

Graph of

Graph of

1. From the left (red arrows):

   • As approaches from the left, approaches .

   • Which direction do you evaluate the limit of from? The input to is . This means that the -axis of the graph is .

     We evaluated from the left. From the left, is less than . A lower value of corresponds to the left side of the graph.

     Therefore, we evaluate the limit of from the left side of the jump discontinuity, which is .

2. From the right (green arrows):
   • As approaches from the right, approaches .
   • This is then the input to , which approaches from the right.

To aid visualization for the directions, we can show different points on the and graphs that correspond to the various limits:

Graph of

Graph of

Since the limits from both sides agree, we can say that the limit of as approaches is :

Example Problem: Limits with Undefined Internal Limits

Using the same functions and , evaluate the following limit:

In this case, we are evaluating the limit of as approaches . Look at the graph for - there is a discontinuity at , and the function approaches different values from the left and the right.

Just like before, consider the limit from both sides:

Graph of

Graph of

1. From the left:
   • As approaches from the left, approaches (yellow point).
   • Since is lower to the left of , we evaluate the limit of from the left side of the jump discontinuity, which is .
2. From the right:
   • As approaches from the right, approaches (purple point).
   • As is lower to the right of , we evaluate the limit of from the LEFT side of the jump discontinuity! Look at how the green and purple points correspond in the graph.
   • Therefore, the limit of as approaches from the right is .

Since the limits from both sides are different, the limit of as approaches does not exist. However, we can write the one-sided limits:

Example Problem: Limits with Undefined External Limits

Using the same functions and , evaluate the following limit:

This time, does have a limit as approaches ; it approaches . However, has a jump discontinuity at . As always, consider the limit from both sides:

Graph of

Graph of

1. From the left:
   • As approaches from the left, approaches (red arrow).
   • Since is lower to the left of , we evaluate the limit of from the left side of the jump discontinuity, which is .
2. From the right:
   • As approaches from the right, approaches (green arrow).
   • Since is higher to the right of , we evaluate the limit of from the right side of the jump discontinuity, which is .

Since the limits from both sides are different, the limit of as approaches does not exist. We can write the one-sided limits:

Summary and Next Steps

In this section, we discussed some algebraic properties of limits that can help us evaluate the combined limits of functions.

Here are the key points to remember:

1. The limit of a sum of functions is the sum of their limits.
2. The limit of a product of functions is the product of their limits.
3. The limit of a power of a function is the power of the limit of the function.
4. The limit of a composite function can be evaluated by breaking it down into the limits of the individual functions, provided the conditions for the limit of composite functions are met:
   • The limit of the inner function exists.
   • The outer function is continuous at the limit of the inner function.
5. If the conditions for the limit of composite functions are not met, we may need to consider the limit from both sides and see if they agree.

These properties can be useful when evaluating limits of more complex functions that can be broken down into simpler functions.

Next, now that the preliminaries are out of the way, we can start evaluating limits algebraically, first by direct substitution.