The title might sound daunting, but properties of limits (also called limit laws) are just shortcuts to finding limits of functions.

## How To Use Properties of Limits

To find a limit using the properties of limits rule:

- Figure out what kind of function you are dealing with in the list of “Function Types” below (for example, an exponential function or a logarithmic function),
- Click on the function name to skip to the correct rule,
- Substitute your specific function into the rule.

## Function Types

**Click a function name in the left column** to skip to that rule.

Rule Name | Notation | Example |

1. Constant function | f(x) = C | y = 5 |

2. Constant multiplied by another function | k * f(x) |
5 * 10x^{2} |

3. Sum of functions | f(x) + g(x) + … | 10x^{2} + 5x |

4. Product of two or more functions | f(x) * g(x) * … | 10x^{2} * 5x |

5.Quotient Law | f(x) / g(x) | 5x / 10x^{2} |

6. Power functions | f(x) = ax^{p} |
10x^{2} |

7. Exponential functions | f(x) = b^{x} |
10^{x} |

8. Logarithmic functions | f(x) = log_{b}x |
log_{10}x |

## 1. Constant Function

The limit of a constant function **C** is equal to the constant.

**Example**: if the function is y = 5, then the limit is 5.

## 2. Constant Multiplied by a Function (Constant Multiple Rule)

The limit of a constant (*k*) multiplied by a function equals the constant multiplied by the limit of the function.

**Example**: Find the limit of f(x) = 5 * 10x

^{2}as x→2.

- The limit of f(x) = 5 is 5 (from rule 1 above).
- The limit of 10x
^{2}at x = 2 can be found with direct substitution (where you just plug in the x-value): 10((2^{2}) = 40 - Multiply your answers from (1) and (2) together: 5 * 40 = 200

**Tip**: Plot a graph (using a graphing calculator) to check your answers.

## 3. Sum of Functions

The limit of a sum equals the sum of the limits. In other words, figure out the limit for each piece, then add them together.

For step by step examples, see: Sum rule for limits.

## 4. Product of Two or More Functions

The limit of a product (multiplication) is equal to the product of the limits. In other words, find the limits of the individual parts and then multiply those together.

**Example**: Find the limit as x→2 for x^{2} · 5 · 10x

- The limit of x
^{2}as x→2 (using direct substitution) is x^{2}= 2^{2}= 4 - The limit of the constant 5 (rule 1 above) is 5
- Limit of 10x (using direct substitution again) = 10(2) = 20
- Multiply (1), (2) and (3) together: 4 · 5 · 20 = 400

## Extended Product Rule

Any “extended” formulas in properties of limits are just extensions of other formulas. This one is just an extension of the product rule above: you can just keep on multiplying as many parts as you need (e.g. a * b * c * d * …).

## 5. Quotient of Two or More Functions

The limit of a quotient is equal to the quotient of the limits. In other words:

- Find the limit for the numerator,
- Find the limit for the denominator,
- Divide the two (assuming that the denominator isn’t zero!).

## 6. Power Functions

The rule for power functions states: The limit of the power of a function is the power of the limit of the function, where p is any real number.

**Example**: Find the limit of the function f(x) = x^{2} as x→2.

- Remove the power: f(x) = x
- Find the limit of step 1 at the given x-value (x→2): the limit of f(x) = 2 at x = 2 is 2. You can use direct substitution or a graph like the one on the left.
- Put the power back in: 2
^{2}= 4

A particular case involving a radical:

Also, if f(x) = x^{n}, then:

This particular part of the properties of limits “rule” for power functions is really just a shortcut: The limit of x power is *a* power when x approaches *a*.

## 7. Exponential Functions

## 8. Logarithmic Functions

## Properties of Limits: References

Gunnels, P. (undated). Limit Laws. Retrieved May 29, 2019 from: http://people.math.umass.edu/~gunnells/teaching/Sample_Lecture_Notes.pdf