Limits (Old Notes)

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Limits are a fundamental concept in calculus. They are used to define the derivative and the integral, and are used to define continuity.

In essence, a limit asks the question: "What value does a function approach as it gets closer and closer to a certain point?"

Table of Contents

• Table of Contents
• Notation
• Examples
• One-Sided Limits
• Estimating Limits
  • Graph Visualization
  • Tables
• General Procedure
  • Trig Identity Example
  • Factoring Example
  • Rationalizing Example
• Combining Functions
  • Sum and Difference
  • Products
  • Composition
• Squeeze Theorem
  • Squeeze Theorem Example 1
• (\varepsilon-\delta) Definition of Limits
  • Example Problem: Constructing an Epsilon-Delta Proof
• Continuity
  • Continuity Over an Interval
  • Example Problem: Removing Discontinuities
• Limits at Infinity
  • Definition
  • Simple Example
  • Exponents
• Intermediate Value Theorem

Notation

The limit of a function as approaches is denoted by:

This is read as "the limit of as approaches ." It asks, "what value does approach as gets closer and closer to ?"

Examples

For instance, consider the function . As gets closer and closer to 0, gets larger and larger. This can also be viewed on a graph:

Notice how while it is impossible to actually reach 0, the function gets larger and larger as gets closer and closer to 0. This is why the limit of as approaches 0 is .

Written mathematically:

One-Sided Limits

Let's say we want to get the limit of a function. This time, we're not going to get the function itself, but rather just the graph.

If we try to approach , you see that there's a massive (infinite) gap between the two sides. And you approach different values depending on which side you're coming from.

This is why we have the concept of one-sided limits.

The limit of as approaches from the left is denoted by:

And similarly, the limit of as approaches from the right is denoted by:

In this case, using the graph alone, we can see that:

Estimating Limits

There are many ways to estimate a limit. In fact, we just approximated one by looking at the graph above.

Graph Visualization

Graphing the function and seeing what value it approaches as gets closer and closer to the point in question is a good way to estimate a limit.

Example 1

Before anything, we can try to evaluate the limit by naively plugging in :

Oh no! We can't divide by 0! This is a problem. However, we can still try to estimate the limit by graphing the function:

Notice that while there's a gap at , the function seems to approach 1 as gets closer and closer to 0. This is why:

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This is not exactly why, but remember that we are building qualitative intuition here. The actual proof is more rigorous and involves the squeeze theorem, which we will cover later.

Tables

Another way to estimate a limit is by creating a table of values. This is especially useful when the function is not graphable, or you don't have access to a graphing calculator.

Basic Example

Let's say this table is given:

We can estimate the limit of as approaches 4 by looking at the values of as gets closer and closer to 4.

Generally, look for these things when constructing or analyzing a table like this:

• The values chosen for should get closer and closer to the value in question.
• The values of should be getting closer and closer to a single value.
• If the values of are getting closer and closer to a single value, then that value is our estimate for the limit.
• The limit can be two-sided, and that's shown by the values of getting closer and closer to a single value from both sides.

Double-Sided Tables

Let's say we want to find the limit of as approaches 1. Looking at the table though, it seems like there's a gap between and . This means that there is likely no limit.

However, if we look at the values of as gets closer and closer to 1 from the left, we see that the values of are getting closer and closer to 10. Likewise, if we look at the values of as gets closer and closer to 1 from the right, we see that the values of are getting closer and closer to 7.

General Procedure

For this limit:

You can apply this general procedure:

1. Try to substitute in . If you get a number, then that's probably the limit.
2. If you get where , then you probably hit an asymptote.
3. If you get , then you got an indeterminate form.
4. If it's indeterminate, try to rewrite the limit by factoring, rationalizing, or using trigonometric identities.
5. If all else fails, try to estimate the limit using a graph or a table.

Trig Identity Example

The function is defined as follows:

Find the limit of as approaches .

Let's first try to substitute in :

We got an indeterminate form. This means we should rewrite the limit. In this case, the numerator, , can be rewritten using the double angle identity:

So the limit becomes:

Factoring Example

The function is defined as follows:

Find the limit of as approaches 2.

As always, let's first try to substitute in :

Once again, we got an indeterminate form. This means we should rewrite the limit. In this case, the numerator, , can be factored:

So the limit becomes:

Note that this new function works for because if , then we are dividing by .

However, it allows us to understand what happens for values close to , and as such, we can calculate the limit.

More formal please!

The more formal way to say this is:

Let for all . Then:

Rationalizing Example

The function is defined as follows:

Find the limit of as approaches 2.

This also gives us an indeterminate form when we try to substitute in . We can rewrite the limit by rationalizing the numerator through multiplying by the conjugate:

While this new function works for , it allows us to understand what happens for values close to , and as such, we can calculate the limit.

Combining Functions

If you have a limit of a function that's a combination of other functions, you can use the limit laws to simplify the limit.

Sum and Difference

The limit of sum is the sum of limits.

Products

The limit of a product is the product of limits.

Composition

This is a bit more interesting. Let's say you want to find the limit of as approaches . This is called a composition of functions, where you first apply and then apply to the result.

For this to work, two conditions must be met:

1. A value must exist such that .
2. The function must be continuous at .

If these conditions are met, then:

Squeeze Theorem

The squeeze theorem is a powerful tool for finding limits. It's especially useful when you have a function that's hard to evaluate directly.

Let's use an analogy: Alice, Bob, and Charlie all have different heights. Bob is taller than Alice, and Charlie is taller than Bob.

Mathematically: .

Let's say we know Alice's height to be and Charlie's height to also be . Since Bob's height must be between Alice's and Charlie's, we can conclude that Bob's height is also .

This is the essence of the squeeze theorem. If you have a function that's always between two other functions and , and the limits of and as approaches are the same, then the limit of as approaches is also the same.

Squeeze Theorem Example 1

Remember our function ? We estimated the limit to be 1 by graphing the function. However, we can also use the squeeze theorem to rigorously prove that the limit is 1.

As we know, trigonomic functions can be modeled by the unit circle. We will use this to visualize this proof.

Firstly, this is the unit circle:

The function can be visualized as the height of the small, blue triangle. If we extend the line to the right and draw a line from the right to the top, we get a new triangle.

The tangent of the angle would be the opposite over the adjacent, which is . Therefore, the height of the triangle is .

Now we are going to adjust our blue triangle to touch the unit circle on the right.

With this in mind, simply looking at the unit circle, we can see that the red triangle is bigger than the blue triangle.

Finally, let's consider the area of the unit circle sector.

The area of the sector can be expressed as a fraction of the area of the whole circle:

And looking at the unit circle, we can see that the area of the sector is bigger than our original blue triangle.

Combining the two inequalities and performing some algebraic manipulation, we get:

Now let's consider our limit again. Since we just want to find out when the angle approaches , we only need to consider the first and fourth quadrants:

• In the first quadrant, is positive and is positive.
• In the fourth quadrant, is negative and is negative.
• In both quadrants, is positive.

As such, we can drop the absolute value signs:

This can be seen on the graph as well:

The red line represents . Notice how it is always sandwiched between the other two lines. And more crucially, that they all seem to approach the same value as approaches . To confirm, let's take the limit of all three sides as approaches :

And since the limit of as approaches is 1, we can conclude that:

That's it. We've proven that the limit of as approaches 0 is 1.

- Definition of Limits

Our definition of limits is simple and intuitive, but it's not rigorous. We can't just say "the function gets closer and closer to a value" - everything in Mathematics needs to be defined precisely.

One way to define limits rigorously is using the - definition of limits, or the "epsilon-delta" definition of limits.

Graphically, this is what it looks like:

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Suppose we want to find out what this function approaches as approaches . We've already done this by looking at values around . In this case, we have a range of values around shown by the red lines, and a range of values around shown by the blue lines. The - definition of limits tells us that if we decrease the size of the range of values, the size of the range of values also decreases.

Essentially, you're saying that we can get as close as we want to by getting close enough to . For example, if the limit exists, we can get within of by getting within some distance of .

This distance for the output is denoted by the symbol (epsilon), and the distance for the input is denoted by the symbol (delta).

Therefore, you can state that, if the limit exists at , then if you want to be within of , you can find a such that if is within of , then is within of .

This can be written mathematically as well; you can rewrite the delta part as , and the epsilon part as . Finally, we can generalize this to any point . Then, the full definition of the limit is:

Example Problem: Constructing an Epsilon-Delta Proof

The function is defined as follows:

Using the - definition of limits, prove that .

The general procedure for constructing an - proof is to either:

• Define , then show that for any you can find a that satisfies the definition.
• Define as a function of .

So firstly let's construct a graph to visualize the function:

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Let's consider inputs of that are away from (but not equal to ).

Recall that the definition of the limit requires two conditions:

1. .
2. .

Since the definition requires , we can plug in our values to obtain .

Now, we're going to try to manipulate this to make it look like the second condition, .

Our proposed limit is , and for . Therefore, we need to make it look something like .

We can start with the delta part, and then multiply by :

Now the left-side is the same. We need to make the right-side look be , i.e. define as a function of .

So we can just make :

As such:

• For any ,
• As long as you set ,
• Then any that is within of will be within of .

Therefore, we have proven that .

Continuity

Conceptually, a function is continuous if you can draw it without lifting your pen. This means that the function doesn't have any jumps or holes.

Formally, a function is continuous at if:

1. is defined.
2. exists.
3. .

For example:

Provided that this function is continuous for all ,

Counterexample:

Provided that this function is not continuous at ,

Hence, knowing whether a function is continuous at a point is important for understanding the behavior of the function.

Continuity Over an Interval

A function is continuous over an interval if it is continuous at every point in that interval.

Therefore, a function is continuous over an interval if:

1. is defined for all .
2. exists for all .
3. for all .

If a function is continuous for every , then it is continuous on all real numbers.

Example Problem: Removing Discontinuities

The function is defined as follows:

This function is not continuous at . Find the value of that would make continuous at .

To make continuous at , we need to find the value of that would make the limit of as approaches equal to . One way to do this is through factoring:

This new function works for , but it allows us to understand what happens for values close to , and as such, we can calculate the limit.

Therefore, the value of that would make continuous at is . We can rewrite as:

Limits at Infinity

Limits at infinity are used to describe the behavior of a function as gets larger and larger.

Definition

The limit of as approaches is denoted by:

AKA, "what value does approach as gets larger and larger?"

Simple Example

This is because as gets larger and larger, gets smaller and smaller. As such, the limit of as approaches is 0.

Exponents

It's often useful to think of limits at infinity in terms of exponents. In a fraction, it's useful to look for the highest power of in the numerator and the denominator, because that tells us how fast they grow.

For instance, consider the function:

As gets larger and larger, the grows the fastest. The , , and matter less and less. So while both the numerator and the denominator grow as gets larger, the numerator grows faster, and the entire fraction increases. Hence:

If the highest power of in the numerator and the denominator are the same:

If you think about it, as gets larger and larger, and increase, causing and to matter less. Therefore, we can ignore them for now.

While the numerator and denominator grows, they grow at the same rate; will always be of . Therefore:

In general, you look for the coefficients of the highest powers.

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Another way to think of it is,

(This is purely conceptual and not mathematically precise, but is nevertheless a helpful intuition)

So, the ratio of coefficients is , which is . This is purely metaphorical and shouldn't be taken as a mathematical proof or be used during a test.

Intermediate Value Theorem

The Intermediate Value Theorem (IVT) states something that seems very obvious:

If you have a continuous function that is continuous at every point in the interval , then for every value between and , there exists a value between and such that .

Or, more informally:

If you have a continuous function that starts at one value and ends at another, then it must take on every value in between.

It sounds really obvious, but it's actually a very powerful theorem that reveals a lot about continuity.

The proof, however, is not as obvious. It involves creating a new function and showing that and have opposite signs, which means that must cross the -axis at some point. Proofs like this are often dealt with in Real Analysis courses. For now we're just interested in the intuition behind the theorem, as well as applying it to solve problems.