Chapters

## What is a Limit?

When we want to find the limit of something, we are generally trying to find the limit of a function. You can think of a limit as the **boundary** of a function. Take a look at the following example.

x | y |

1 | 0.0625 |

2 | 0.003906 |

3 | 0.000244 |

... | ... |

8 | 2.33E-10 |

When we plug in **bigger** and **bigger** values into x for this function, we can see it gets closer and closer to zero.

## Limit Notation

When we are asked to find the limit of a function, there is a certain notation we **encounter.** Let’s take a look at this notation.

A | Symbol for the limit |

B | As the inputs, x, approach a specific value a |

C | The function we want to find the limit for |

The notation for approaching the value **a** on the right and left can be found below.

A | B |

Approach from right side | Approach from left side |

## Approach from the Left

Sometimes, the function at the specified value **a** doesn’t exist. Take a look at some **examples** below.

Function | Results is Unknown |

When this happens, we can instead get as close to the specific value a as **possible.** Take a look.

x | y |

0.8 | 1.8000 |

0.9 | 1.9000 |

0.99 | 1.9900 |

0.999 | 1.9990 |

0.9999 | 1.9999 |

As we approach 1, we see that our value gets closer and closer to 2. We can say that our **limit** is 2.

## Approach from the Right

In the previous section, we wanted to approach 1. So, we plugged in values that were closer and closer to 1. In that example, we approached 1 from the **left side**. Take a look at why:

We can also **approach** our a value from the right side. Take a look at the result.

x | y |

1.1 | 2.1000 |

1.01 | 2.0100 |

1.001 | 2.0010 |

1.0001 | 2.0001 |

As you can see, we get the same **result:** as we approach 1 form the right side, we get closer and closer to two.

## Indeterminate Forms

Indeterminate forms have unknown values. This means we do know their true value, whether they are **undefined** or simply unknown. We can typically divide common indeterminate forms into three groups.

A | B | C |

Fraction | Standard | Power |

## Problem 1

In this problem, we will review what you have learned about **indeterminate** forms. Given the following table, label each limit that will result an indeterminate form as a approaches 0.

1 | |

2 | |

3 |

## Problem 2

Take a look at the following limit problem.

Taking what you’ve learned about **approaching** limits, approach the value **a** from the right. Next, approach the value a from the left.

## Problem 3

Taking what you know about finding the limit, lets try to find the limit of a function as it approaches **infinity.**

Keep in mind that for this problem you might have to approach the **limit** form the right or left, depending on what result you get.

## Problem 4

In this problem, we will put together some more **knowledge** on indeterminate forms. Take the following function.

Find the limit of the function and then **graph** the function as it approaches its limit.

## Solution Problem 1

In this problem, you were asked to review what you learned about indeterminate forms. Given table, you had to label each limit that will result an indeterminate form as a approaches 0.

1 | Indeterminate form | ||

2 | Indeterminate form | ||

3 | Regular |

## Solution Problem 2

In this problem, you were asked to look at the limit problem and approach the value a from the **right** and from the **left.**

First, we plug in values that approach **2** from the left.

x | y |

1 | 6 |

1.5 | 4.666667 |

1.9 | 4.105263 |

1.99 | 4.01005 |

1.999 | 4.001001 |

Here, we can see the y value gets closer and closer to **4**.

Let’s approach the value 2 from the **right** now and see if the same **pattern** occurs.

x | y |

2.001 | 3.999 |

2.01 | 3.990 |

2 | 4.000 |

2.1 | 3.905 |

2.5 | 3.600 |

Here, we’re also getting **closer** to 4.

We can see the approach from the **right** side.

## Solution Problem 3

In this problem, you were asked to take what you learned about finding the limit and use it to try to find the limit of a function as it approaches infinity. In this problem, we didn’t **specify** whether you had to approach the limit form the right or left.

Let’s first **plug** infinity into the problem.

Here, we **encounter** a problem: infinity is endless and therefore unknown/undefined. However, we can approximate infinity by plugging in higher and higher values into this term.

x | y |

10 | 0.1 |

100 | 0.01 |

1000 | 0.001 |

90000 | 1.11E-05 |

As you can see, when we approach higher and higher values of x, we get closer and closer to **zero.**

Because this term is approximately zero, we can say that the limit is **approximately** 2.

## Solution Problem 4

In this problem, we were asked to find the limit of the function and then graph the function as it approaches its **limit.** We can already see that plugging in this a value will result in an indeterminate form.

However, we can try plugging in values that approach zero into the **x** place.

x | y |

1 | 1 |

0.5 | 2 |

0.01 | 100 |

0.001 | 1000 |

0.0001 | 10000 |

We can see that as we plug in smaller and smaller values, the y value approaches bigger and bigger numbers. We can also see this **visually:**

Without using **l’Hopital’s** rule for finding the limit of indeterminate forms, we can say that as x approaches zero, the limit approaches infinity.

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