Calculus How To

Properties of Limits (Limit Laws)

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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:

  1. 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),
  2. Click on the function name to skip to the correct rule,
  3. 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 * 10x2
3. Sum of functions f(x) + g(x) + … 10x2 + 5x
4. Product of two or more functions f(x) * g(x) * … 10x2 * 5x
5.Quotient Law f(x) / g(x) 5x / 10x2
6. Power functions f(x) = axp 10x2
7. Exponential functions f(x) = bx 10x
8. Logarithmic functions f(x) = logbx log10x

1. Constant Function

constant ruleThe 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)

Constant Multiplied by a FunctionThe limit of a constant (k) multiplied by a function equals the constant multiplied by the limit of the function.

Limit of 5 * 10x2 as x approaches 2.

Example: Find the limit of f(x) = 5 * 10x2 as x→2.

  1. The limit of f(x) = 5 is 5 (from rule 1 above).
  2. The limit of 10x2 at x = 2 can be found with direct substitution (where you just plug in the x-value): 10((22) = 40
  3. 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

sum of functions limit rule

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

limit 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 x2 · 5 · 10x

  1. The limit of x2 as x→2 (using direct substitution) is x2 = 22 = 4
  2. The limit of the constant 5 (rule 1 above) is 5
  3. Limit of 10x (using direct substitution again) = 10(2) = 20
  4. Multiply (1), (2) and (3) together: 4 · 5 · 20 = 400

Extended Product Rule

properties of limits

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

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

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.

properties of limits powerExample: Find the limit of the function f(x) = x2 as x→2.

  1. Remove the power: f(x) = x
  2. 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.
  3. Put the power back in: 22 = 4


A particular case involving a radical:
Power Functions real number


Also, if f(x) = xn, then:
Power Functions limit

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

Exponential Functions limit

8. Logarithmic Functions

Logarithmic Functions limit

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

CITE THIS AS:
Stephanie Glen. "Properties of Limits (Limit Laws)" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/limit-of-functions/properties-of-limits-laws/
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