BC Calculus KA: Limit Properties

The Limit of [f(x) + g(x)] as x approaches A…

• [limit of f(x)] + [limit of g(x)] as x approaches A

• sum of the two limit values

Sum Rule of limits

• The limit of the sum of two functions is equal to the sum of the individual limit values as x approaches a common point

The Limit of [f(x) - g(x)] as x approaches A…

• [limit of f(x)] - [limit of g(x)] as x approaches A

• difference of limit values

Difference Rule of limits

• The limit of the difference between two functions is equal to the difference of the individual limit values as x approaches a common point

Product Rule of limits

• The limit of the product of two functions is equal to the product of each individual limit value as x approaches a common point

Constant Multiple Property of Limits (If a constant is multiplied by a limit)

• The product is equal to the constant times the individual limit value

Quotient Rule of Limits

• The limit of (a function divided by another function) is equal to the quotient of the individual limit values, as x approaches a common point

Constant Division Property of Limits (If a limit is divided by a constant)

• The quotient is equal to the individual limit value divided by the constant

Quotient Rule of Limits: If the limit of the divisor function is equal to 0 as x approaches a common point

• The quotient is undefined

• Therefore, the limit does not exist

Exponent Rule of Limits

• The limit of a function raised to an exponential power, as x approaches a given point, is equal to the limit value raised to the exponential power

• Applies to all types of exponents: whole number, negative integer, fraction

Sum of two limits where one or both does not exist

• If the sum of…

  • the values approached by both limits from values greater than x

  • and

  • the values approached by limits from values less than x

• are equal…

• Then the sum of the two limits as x approaches a given point is equal to that sum (of greater and lesser individually)

Product of two limits where one or both does not exist

• If the product of…

  • the values approached by both limits from values greater than x

  • and

  • the values approached by limits from values less than x

• are equal…

• Then the product of the two limits as x approaches a given point is equal to that product (of greater and lesser individually)