Chapters

## What is a Limit?

Limits are one of the basic **building blocks** of calculus - however, it is sometimes hard to understand their properties. In order to explore the properties of limits, let’s define what a limit is.

When we talk about limits, we usually are dealing with functions. A limit is defined as some value that a function approaches when the input of that function reaches a specific value. Let’s break this down: a function takes a number as an **input** and gives a number as an **output.**

As you can see, when the input approaches higher and higher values, the output gets closer and **closer** to zero. This is an example of the limit of a function.

## Approaching 1

When we talk about **‘approaching’** a value in math, we mean that we never actually get to that value. Let’s take the following function.

Let’s try to set **x = 1**.

This result is actually a special value known as the indeterminate form, which means that we don’t actually know the true value of them. Because the function at x = 1 is **unknown,** we can instead try approaching 1.

x | y |

0.8 | 2.44000 |

0.9 | 2.71000 |

0.99 | 2.97010 |

0.999 | 2.99700 |

0.9999 | 2.99970 |

0.99999 | 2.99997 |

This is called **approaching** the limit from the ‘left,’ which is easier to understand by looking at a number line.

We can see that when x approaches 1, it gets close to 3, which is the **limit.**

## Approaching Infinity

One common task you will see when working with limits is being asked to approach **infinity.** Infinity, however, is defined as an **endless** number. How can we approach, or calculate, something that is endless?

10 |

10,000 |

1,000,000 |

... |

We can apply the same method we used in the previous example. Take a look at this **common function:**

If we set x = infinity, we would get another undefined, or unknown, value. However, we can approach infinity by plugging in **bigger** and **bigger** values.

x | y |

10 | 0.1 |

100 | 0.01 |

1000 | 0.001 |

10000 | 0.0001 |

100000 | 0.00001 |

1000000 | 0.000001 |

When we graph this, we can see that as x approaches **infinity,** the output approaches **zero.** This is the limit of this function.

## Limit Notation

So far we have talked about the definition of a limit and some examples. Now, let’s turn to the formal, **mathematical** notation of limits. You will typically see limit notation like this:

a | Value that we’re approaching |

f(x) | Function |

c | The limit |

lim | Limit symbol |

Recall that we talked about approaching a limit from the ‘left’ or ‘right’ side. This **distinction** is written as the following.

+a | Approach from the right |

-a | Approach from the left |

Verbally, we say that as x approaches a, then the function **f(x)** approaches the value c. Let’s take the example in the previous section, where we worked with infinity.

Here, we would say that as x approaches infinity, the function **approaches** 0.

## Limits of a Function

There are several different ways you can find the limit of a function. The easiest and most **natural** method is to simply substitute the value a as the input. Let’s take an example.

Previously, when we substituted infinity into the function , we had an undefined value. Here, we can **substitute** 10 into the function without any issues because it is defined.

When we have an undefined value or indeterminate form, what should we do? There are a couple of **different** ways you can do this, described in the table below.

Method | When to Use | |

Method 2 | Simplify the function | When plugging in the value a results in an indeterminate form |

Method 3 | L’Hopital’s Rule | Indeterminate forms like or |

## Continuity

Continuity is an important concept when it comes to functions and limits. A function is **continuous** if there are no breaks of any kind on the graph. Let’s take a look at some examples of non-continuous functions.

A | B | C |

Vertical asymptote | Break | Jump |

## Break

When there are breaks in the graph of any kind, many important rules and methods can’t be applied to a **function** at that point. Let’s take a look at the formal definition of continuity.

A | B |

If f(c) exists | If the limit at x=c exists |

## Properties of Limits

In order to find the limit of a function, there are several rules you can use in order to make finding the **limit** easier.

### Sum Rule

The sum rule states that the limit as x approaches a value a of the sum of **two functions,** f(x) + g(x), is the same thing as the limit as x approaches value a of f(x) plus the limit as x approaches value a of g(x).

The limit of the sum of two functions = The sum of their limits |

### Constant Rule

The constant rule involves constant functions. A constant function is **any function** that is one constant c.

The limit of a constant function = constant |

### Product Rule

The product **rule** states as x approaches a value a of the product of two functions, f(x) * g(x), is the same thing as the limit as x approaches value a of f(x) multiplied by the limit as x approaches value a of g(x).

The limit of the product of two functions = The product of their limits |

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