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.

 

power_function

 

x y
1 0.0625
2 0.003906
3 0.000244
... ...
8 2.33E-10

 

log_functions

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

 

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Let's go

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.

 

limit_notation_function

 

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.

 

limit_approach

 

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
limlimits_{x to 3} ; frac{1}{x-3} frac{1}{3-3} = frac{1}{0}
limlimits_{x to 0} ; (1+8x)^{frac{1}{x}} = 1^{infty}
limlimits_{x to 1} ; frac{x^{2} - 1}{x - 1} = frac{0}{0}

 

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

 

rational_function

 

 

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:

 

approach_limit

 

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.

 

indeterminate_forms

 

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 frac{1}{f(x)}
2 frac{x^2}{x}
3 frac{3x + 1}{5}

 

Problem 2

Take a look at the following limit problem.

 

limit_function_rational

 

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.

 

rational_function_limit_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.

 

function_limit

 

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.

 

limit_approaching_zero

1 frac{1}{f(x)} frac{1}{0} Indeterminate form
2 frac{x^2}{x} frac{0}{0} Indeterminate form
3 frac{3x + 1}{5} frac{1}{5} 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.

 

limit_graph

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.

 

limit_approaching_value

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.

 

limit_approaching_infinity

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 frac{1}{x} 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.

 

approaching_zero

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

 

approaching_limit

 

 

 

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.

limit_approaching_zero_function

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:

 

approaching_infinity

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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Danica

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