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.
Click a function name in the left column to skip to that rule.
|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|
The limit of a constant (k) multiplied by a function equals the constant multiplied by the limit of the function.
- The limit of f(x) = 5 is 5 (from rule 1 above).
- The limit of 10x2 at x = 2 can be found with direct substitution (where you just plug in the x-value): 10((22) = 40
- Multiply your answers from (1) and (2) together: 5 * 40 = 200
Tip: Plot a graph (using a graphing calculator) to check your answers.
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.
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
- The limit of x2 as x→2 (using direct substitution) is x2 = 22 = 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 * …).
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!).
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) = x2 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: 22 = 4
A particular case involving a radical:
Also, if f(x) = xn, 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.
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
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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