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.

function_input

 

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.

 

exponential_function

 

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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.

rational_function

Let’s try to set x = 1.

Indeterminate_form

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.

limit_approach_left

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
...
\infty

 

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

function_indeterminate

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.

exponential_decrease

 

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:

limit_notation

 

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.

limit_approach_left_and_right

 

+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.

limit_function

Here, we would say that as x approaches infinity, the function \frac{1}{f(x)} 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 \frac{1}{f(x)}, we had an undefined value. Here, we can substitute 10 into the function without any issues because it is defined.

finding_limit_example

 

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 \frac{0}{0} or \frac{\infty}{\infty}

 

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.

 

non-continuous_funtion

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.

 

l'hopital_conditions

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).

 

\lim\limits_{x \to a} \; f(x) + g(x) = \lim\limits_{x \to a} \; f(x) +\lim\limits_{x \to a} \; 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.

 

\lim\limits_{x \to a} \; C = 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).

 

\lim\limits_{x \to a} \; f(x) * g(x) = \lim\limits_{x \to a} \; f(x) *\lim\limits_{x \to a} \; g(x) The limit of the product of two functions = The product of their limits

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Danica

Located in Prague and studying to become a Statistician, I enjoy reading, writing, and exploring new places.