The

**limit of functions**refers to the output (i.e. y-value) that a given function intends to reach as “x” moves towards some value. It answers the question “Which number did this function get to?” as well as “Which number did this function

*try*to get to?”.

In real life, driving “at the speed limit” might mean you’re going at exactly 70 mph. But it can also mean 69.998, 69.999 mph. If you get very, very close, you can still say you drove at the speed limit; It’s the same thing in calculus: you’re looking for that intended value (e.g. 70 mph),

**even if you don’t quite get there**.

**Limit of Functions: Contents (Click to go to that article):**

## Definitions and Properties

- What is a Limit?
- One Sided Limits
- Limit from Below
- Limit from Above
- When Does the Limit Not Exist?
- What are Indeterminate Limits?
- Properties of Limits
- Slope of a Tangent Line using the Definition of a Limit

## Finding Limit of Functions: Techniques

- Direct Substitution.
- The Squeeze Theorem
- Find Limits Using The Formal Definition of a Limit
- Find Limits
**Graphically** - Find Limits
**Numerically** - L’Hospital’s rule

## Find Limit of Functions for Specific Situations:

- Polynomial Functions
- Linear Functions
- How to Find the Limit of a Secant Function
- Limit of Sum & Difference
- Finding Limits of
**Upper and Lower Sums** - Limits at infinity
- Limit of Product/Quotient

## Technology

## What is a Limit?

A **limit **is a number that a function approaches.

For example, take the function f(x) = x + 4. If you plug x = 5, the function equals:

f(5) = 5 + 4 = 9.

That number, 9, is the **limit **for this function at x = 5.

If you take a look at the graph of f(x) = x + 4, you’ll see that all of the numbers surrounding x = 5 (for example, 4.9, 4.999 or 5.1) all get close to (i.e. approach) the limit of 9.

Another way of thinking of it, is that the output (y-value) function tops out at that particular value around that point (x = 5).

## Notation

Limit notation is just used as **shorthand**. Instead of saying “the limit at x = 5 approaches 9 for the function f(x) = x + 4”, you would write:

Where the arrow just means “approaches.”

The general way of writing the notation, without reference to any specific function, is:

What the formula is basically telling you is that you plug in some number “a” into the function, and you get out some number L, the limit of the function.

You might also see the notation written like this:

This doesn’t mean that the *limit *is infinity. It means that you’re plugging in **larger and larger x-values** (i.e. x-values that are getting closer and closer to infinity) to see what happens.

Limits answer the question “Which number did this function get to?” as well as “Which number did this function *try *to get to?”. In other words, a function might try to get to a certain point, but it doesn’t actually get there. This makes is very useful for working with functions that have holes or gaps, like this one:

## Limit of Functions: Limit from Below

**Limit from Below**, also known as a *limit from the left*, is a number that the “x” values approach as you move from left to right on the number line. The possible values of *x * approach a chosen value (e.g. x = 4) but never actually reach that value (e.g. 3.9, 3.99, 3.9999…).

In general terms, the limit from below restricts the variable *x* to a value **less than** the number the variable approaches. Again, these potential values of variable *x* are a limitless sequence of rational numbers.

## A Simple Example

As an example, let’s take the variable *x* as it approaches the value 3 from below (or from the left):

x → 3

The x-values will be a sequence of numbers with values less than 3.

2.9, 2.99, 2.999, 2.9999, 2.99999….

While the variable *x* will get increasingly close to the value of 3, *x* will never equal the number selected as the point for the limit.

## Example of Limit from Below

f(x) = x* ^{2}* as x → 3 from below.

Step 1: Choose a series of x-values that are very close to the stated x-value, coming from the left of the number line. For this example, I’ll use the sequence values for *x* from above:

2.9, 2.99, 2.999, 2.9999, 2.99999

Step 2: Enter your x-values into the given function. The given function is x* ^{2}*, so the sequence

*x*will become:

2.9* ^{2}* = 8.41

2.99

*= 8.9401*

^{2}2.999

*= 8.994001*

^{2}2.9999

*= 8.99940001*

^{2}2.99999

*= 8.9999400001*

^{2}At the point x = 3 (coming from the left on the number line), *x ^{2}* approaches 9.

## Solve Limits Step by Step Using The Formal Definition of a Limit

This section shows you how to solve limits using the formal limit definition.

The **formal definition of a limit** is:

“For any ε > 0, there is a δ > 0 so that |f(x) – L| < ε when 0 < |x-a| < δ”.

What it *means* is that when x gets close to a number, f(x) gets close to L, a limit. If you want to find limits, it’s more intuitive to solve limits numerically or solve limits graphically. However, it is possible to solve limits step by step using the formal definition.

## Solve Limits Step by Step Example

*Sample question*: Prove that the limit of f(x) = 2x + 4 is 10 as x approaches 3.

Step 1:** Assign δ and ε** to your x-value and your function, setting them up as inequalities:

0 < |x – 3| < δ

|(2x + 4)-10| < ε

Step 2:** Find a value for δ** (in this example, δ is |x-3|) in terms of ε. Start with the ε portion from step 1:

|(2x + 4) – 10| < ε =

|(2x-6| < ε = (Simplifying)

2|x-3| < ε =

|x-3|< ε/2 =

δ < ε/2

Step 3:** Turn the inequality into an equals sign**:

δ = ε/2

Step 4:** Use algebra to manipulate the δ** from Step 1. This time, your goal is to go from 0 < |x – 3|< δ): to |(2x + 4) – 10| < ε

0 < |x – 3|< δ =

0 < |x – 3|< ε/ 2 = (using the value for delta you derived in Step 2)

0 < 2|x – 3|< ε=

0 < |2x – 6| < ε=

0 <|2x + 4|- 10 < ε

**Tip: **Step 2 is really all guesswork. Look for ways to make the numbers in the ε portion of the equation equal the δ numbers. In this example, there is a “3” in δ, so we simplified to 2x-6 because 6/2=3. In Step 4, the sequence is essentially reversed to prove the formula works.

## Finding Limit of Functions Graphically

It’s almost impossible to find the limit a functions *without* using a graphing calculator, because limits aren’t always apparent until you get very, very close to the x-value. Watch the short video for an introduction to finding limits graphically (without a calculator), or read on below for more information about finding limits graphically with a TI-89.

The TI-89 is one of the best calculators for calculus; if you have another brand (like an HP), the steps will still be the same, but you’ll have to refer to your calculator’s user guide for the specific key strokes.

**Example problem:** Find the limit for the function 3x^{2} – 3 / x^{2} – 9 as x approaches 0

Step 1: **Enter the function** into the y1 slot of the “Y=” window. To open the window, press the Diamond key, then press the F1 key. Type the equation in and then press enter.

Step 2: **Press** the Diamond key, then F2 to enter the Window screen. This screen allows you to tell the calculator what view you want of the graph.

Step 3: **Type a value into the xmin, xmax, ymin and ymax areas.** For many functions, this will be guesswork until you can determine the behavior of the graph. For this particular example, type -5 for xmin, 5 for xmax, -1000 for ymin and 1000 for ymax (we are choosing a small window for the x-values because we want to see the limit around x = 0).

Step 4: Press the Diamond key and then F3 to graph the function.

Step 5: Press F3 for Trace.

Step 6: Press 0 and then press ENTER to see the y-value for x = 0. In this case, it will return nothing—x is undefined at this point.

Step 7: Repeat step 6, entering smaller increments close to x, like .001, .0001 and .00001. You should see the function values in this particular case approaching infinity.

**Tip**: Typically, your xmin and xmax are very small and your ymin and ymax are very large. If you are having trouble viewing the graph, construct a table of values so that you have a better idea of the graph’s behavior.

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## How to Find the Limit of a Function Numerically

The limit of a function is a particular value of x that the function approaches. You can find the limit of a function by graphing it—then the limit usually becomes obvious. But what if you have a function whose graph doesn’t help you? That sometimes happens when the limit is **undefined** at one particular point (you won’t be able to see it at all on a graph—it will appear to be a continuous function) or if the graph keeps on going towards **infinity** (“infinity” isn’t an x-value you can graph!). To find the limit for these functions, you’ll want to find the limit of functions numerically, using a table of values. You *could* make a table of values by hand. However, for most of the functions you’ll be dealing with in calculus, making a table of values by hand is impractical. The solution is to use your **TI-89 graphing calculator.**

**Example question**. What is the limit of 3x^{2} – 3 / x^{2} – 9 as x approaches 0?

Step 1: **Enter the function** into the y1 slot of the “Y=” window. To open the window, press the diamond key, then press the F1 key. Type the equation in and then press enter.

Step 2: **Set the table options.** To set the table options, press the diamond key and then press F4 . Change the settings for the given limit. For this example, set the table to start at -1 (just before the limit) and set the increments (Δ) to 1.

Step 3: **Open the table**: press the diamond and then press F5. Note that the table states “undef” for x = 0 and -9 for x = -1. -9 *could* be the limit, but to be sure, look at smaller increments.

Step 4: **Change the table options to smaller increments. ** For this example, use increments of .1, using the procedure in step 2. Note the limit now looks like it might be -309.

Step 5:** Continue changing the table increments to smaller and smaller values** like hundredths, thousandths or more. When the numbers starts to stabilize around a point, you’ll know the limit. For this particular function, the numbers keep on increasing towards infinity.

**Tips**:

- As you decrease the increments, you’ll also want to change the place where your table starts (move the table to start at a place close to the x-value you are looking at). Otherwise, you’ll find yourself paging through hundreds of table values.
- If you want to check values very close to a number, you can also ask for individual table entries instead of the calculator returning a long list of incremental values. From the table options screen, arrow down to “Independent” and change the input from “Auto” to “Ask.” View the table, press ENTER and then enter an x-value (like -0.000000001).

## How to Find Limit of Sums: Upper and Lower

Upper and lower sums enable you to find the area under a curve. A lower sum is calculated by adding up the areas of inscribed rectangles, while an upper sum is calculated by adding up the areas of circumscribed rectangles.

When the number of rectangles increases to infinity, the upper and lower sums converge on one number: the limit.

## Find Limit of Sums Algebraically

If you want to find limits of upper and lower sums, it’s **incredibly tedious** to perform the process algebraically. Several pages of algebra are required to solve for the upper and lower sums of even the simplest polynomial function—, like y = 3x^{2}. if you want to use algebra, follow the (very lengthy) explanation that Stan Brown provides on his article “Area by Upper and Lower Sums.” You can find that article HERE.

## Find Limit of Sums with the Fundamental Theorem of Calculus

Fortunately, there’s an easier way to find the limit of functions by hand: By using the fundamental theorem of calculus. The fundamental theorem allows you to evaluate definite integrals for functions that have indefinite integrals. The first part of the fundamental theorem states that if you are evaluating indefinite integrals between two points, all you have to do is subtract the value of the integral at the first point from the value of the integral at the second point. If it sounds complicated, it’s actually a cake walk compared to trying to use algebra to find limit of sums! The second part of the fundamental theorem of calculus gives you the indefinite integral for a function.

## Find Limit of Sums on the TI 89

A third option to find limit of sums is the TI89 using the LIMIT command. The LIMIT Command TI 89 guides you through the process, step-by-step.

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## Limit of Functions: Limit from Above (From the Right)

**Limit from above**, also known as

*limit from the right*, is the function

*f(x)*of a real variable

*x*as

*x*decreases in value approaching a specified point

*a*. In other words, if you slide along the x-axis from positive to negative, the limit from the right will be the limit you come across at some point, a.

Possible values for x, as you approach the limit, are a potentially infinite sequence of rational numbers. The concept is represented algebraically as:

But you might also see it written, more specifically, as:

## What is a Limitless Sequence?

In general terms, limit from the right means that the variable *x* will be a value** more than the number the variable approaches**, as the above image illustrates. For example, let’s take the variable *x* as it approaches the value 3 from above (or from the right).

x → 3+

The values of *x* will be a sequence of numbers with values **more **than 3.

3.2, 3.1, 3.01, 3.001, 3.0001 ….

So while the variable *x* will get increasingly close to the value of 3 (what we mean when we say approaching a limit), it will never equal the number selected as a limit.

## Limit of Functions: Example of Limit from Above

In a function, a sequence of inputs (x) will cause a sequence of outputs, *f(x)*. As the input values of *x* approach the *a*, the output values of *f(x)* (i.e. the y-values) approach the limit *A*.

**Example question:** What is the limit from the right of f(x) = x* ^{2}* as x → 3?

Step 1: Choose a few values of x that are very close to the stated x-value (in this case, 3):

3.2, 3.1, 3.01, 3.001, 3.0001.

Step 2: Insert your x-values into the function to get a few y-values:

3.2* ^{2}* = 10.24,

3.1

*= 9.61,*

^{2}3.01

*= 9.0601,*

^{2}3.001

*= 9.006001,*

^{2}3.0001

*= 9.00060001*

^{2}The numbers are getting closer and closer to 9, so you can say that *x ^{2}* will approach 9 as a limit as

*x*approaches 3.

## Limit of Functions: References

Clark, P. L. “Convergence.” 2014. http://math.uga.edu/~pete/convergence.pdf.

Desmos graphing calculator (graph images).

Hight, D. W. A Concept of Limits. New York: Prentice-Hall, 1966.

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