Limits

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Limits of Functions

## Contents

* 1 Basics
* 1.1 What are limits?
* 1.2 Why would anyone need limits?
* 2 Notation
* 3 Properties
* 4 Sample Problem Set #1
* 5 See Also

## Basics[edit | edit source]

### What are limits?[edit | edit source]

Limits are a way to calculate the value that a function approaches. For instance, we could calculate the value of the function f(x) as x approaches 2. Just as easily we can calculate the value of f(x) as x approaches 20, -2, π, 0, or even ∞.

### Why would anyone need limits?[edit | edit source]

There are a number of reasons that someone might want to use limits:

1\. To find the values of functions with asymptotes or missing points
2\. To calculate the slope of a point in calculus
3\. To prove derivatives in calculus

## Notation[edit | edit source]

The notation of a limit function is fairly simple:

lim x -> p f ( x ) = L {\displaystyle \lim _{x\to p}f(x)=L}

This says limit (lim) of f(x) as x approaches p is L.

Usually f(x) is substituted with the contents of the function like so:

lim x -> p x 2 + 2 = L {\displaystyle \lim _{x\to p}x^{2}+2=L}

## Properties[edit | edit source]

lim x -> α ( f ( x ) + g ( x ) ) = lim x -> α f ( x ) + lim x -> α g ( x ) {\displaystyle \lim _{x\to \alpha }(f(x)+g(x))=\lim _{x\to \alpha }f(x)+\lim _{x\to \alpha }g(x)}

lim x -> α ( f ( x ) − g ( x ) ) = lim x -> α f ( x ) − lim x -> α g ( x ) {\displaystyle \lim _{x\to \alpha }(f(x)-g(x))=\lim _{x\to \alpha }f(x)-\lim _{x\to \alpha }g(x)}

lim x -> α ( f ( x ) ⋅ g ( x ) ) = lim x -> α f ( x ) ⋅ lim x -> α g ( x ) {\displaystyle \lim _{x\to \alpha }(f(x)\cdot g(x))=\lim _{x\to \alpha }f(x)\cdot \lim _{x\to \alpha }g(x)}

lim x -> α ( f ( x ) / g ( x ) ) = lim x -> α f ( x ) / lim x -> α g ( x ) {\displaystyle \lim _{x\to \alpha }(f(x)/g(x))=\lim _{x\to \alpha }f(x)/\lim _{x\to \alpha }g(x)}

lim x -> α f ( x ) g ( x ) = lim x -> α f ( x ) lim x -> α g ( x ) {\displaystyle \lim _{x\to \alpha }f(x)^{g(x)}=\lim _{x\to \alpha }f(x)^{\lim _{x\to \alpha }g(x)}}

## Sample Problem Set #1[edit | edit source]

Let's say we have the function  f ( x ) = x 2 {\displaystyle f(x)=x^{2}} . If we want to find the limit as x approaches 4, then:

L = lim x -> 4 x 2 + 2 {\displaystyle L=\lim _{x\to 4}x^{2}+2}

Using two properties of limits:

lim x -> p x + b = lim x -> p x + lim x -> p b {\displaystyle \lim _{x\to p}x+b=\lim _{x\to p}x+\lim _{x\to p}b}

and

lim x -> p x 2 = ( lim x -> p x ) 2 {\displaystyle \lim _{x\to p}x^{2}=(\lim _{x\to p}x)^{2}}

Our problem becomes:

L = ( lim x -> 4 x ) 2 + lim x -> 4 2 {\displaystyle L=(\lim _{x\to 4}x)^{2}+\lim _{x\to 4}2}

If we think about the graph of y=b, then we know that the y value never changes. Which means that at any point on that line, we can expect y to be equal to b. So, for any number b:

lim x -> p b = b {\displaystyle \lim _{x\to p}b=b}

For us, this means that:

lim x -> 4 2 = 2 {\displaystyle \lim _{x\to 4}2=2}

## See Also[edit | edit source]

Wikipedia: Limit of a function