Calculus > Limit Definition

The

**limit**is a calculus term meaning the value that a given function will approach as a variable in the function moves towards some number. Its use is extensive in

**calculus**, and has many real world applications. Taking the below function, you would read it as “The limit of a function of x as x approaches y is z.”

lim f(x)=z

x->y

## Squeeze Theorem and the Limit Definition

A simple illustration of the concept is to **take the limit** of the function y=x+2 as x approaches 5. In other words, we want to know what value of y the function approaches when x is 5. The variable x will need to approach 5 from both sides, unless otherwise noted (because there are two infinities: negative and positive).

Step 1:Take three steadily nearing values for x from both sides. For example, if you are asked to take the limit of x=5, you might choose the numbers 4.7, 4.9. 4.99. 5.01, 5.1 and 5.3. Try to choose numbers that are close to x.

Step 2:Solve for x-values. In this example:

When x = 4.7|4.9|4.99| 5 |5.01|5.1|5.3

f(x) = 6.7|6.9|6.99| 7 |7.01|7.1|7.3

Step 3:Note where the value is trending towards. In this example, y is trending towards 7.

This methodology is known as the **Squeeze Theorem**, or sandwich theorem, which describes the way in which you slowly “squeeze” the value of x towards the intended value. When shown this way, there is an obvious trend towards a value of 7 as x becomes closer to 5 from both directions. Therefore, the **limit of the function** is 7, which might seem like useless additional work given that simply taking the function at x=5 gives the same value.

## A more Advanced Illustration of the Limit Definition

To find a case where a limit can offer a solution where simpler mathematics don’t quite work, take the function (x^2-1)/(x-1). Let’s say we want to find the limit at x=1. If you were to try and make x equal to 1, the function would be **undefined** due to division by 0. If you repeat the same method followed above, you will get the following results:

When x = 0.7|0.9|0.99| 1 |1.01|1.1|1.3

f(x) = 1.7|1.9|1.990|Undef|2.01|2.1|2.3

The value of f(x) as x approaches 1 becomes closer and closer to 2. This is enough proof for an answer.

## Limits Near Infinity

A typical usage of limits is to find a value for numbers that trend towards infinity. For a function of (4x-2)/(x) there would be an undefined value at x = infinity. If you take arbitrarily increasing numbers, you can try to find a pattern to get the limit of the function as follows:

When x = 10000|1000000 |1000000000

f(x) = 3.9998|3.999998|3.999999998

As x approaches infinity, the value of the function steadily moves towards a value of four.

## Advanced Techniques in Taking Limits

One of the advanced tricks in finding **limits approaching infinity** is to discard insignificant figures in the equation. In the equation (4x-2)/(x), we can discard the -2, since infinity outweighs -2 substantially. This leads you to 4x/x, which is 4. While not typically enough proof for a limit, the tip can help you tell if your derived limit is plausible or not.

Reference:

Kouba, Duane. “Limits of Functions Using the Squeeze Principle.” http://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/squeezedirectory/SqueezePrinciple.html