Calculating the limit of product/quotient or sum/differences in math is as simple as bringing the operations outside of the limit function.
The limit of a product is the product of the limits, where they exist. We can write this more formally by saying that:
Almost the same thing is true for quotients. The limit of a quotient is the quotient of the limit, where both limits exist and the limit in the denominator is not equal to zero.
Finding the Limit of Product/Quotient: Worked Examples
Example question: Find the limit:
To solve, start by dividing this up into three separate limits:
Each of these are easy to solve if you know your limit laws. The limit of a constant (lim(4)) is just the constant, and the identity law tells you that the limit of lim(x) as x approaches a is just “a”, so:
The solution is 4 * 3 * 3 = 36.
Note: We don’t need to know all parts of our equation explicitly in order to use the product and quotient rules. It’s enough if we know the limits of the component functions. For example, suppose we know that
And we want to find
Applying the product law, we get 3 x 3 = 9.
The same principle works for the quotient law.
Example problem #2. Find the limit:
Apply the quotient rule, we get 1 divided by the limit of g(x), which is 2. So our answer is 1/2.
Of course, you can always combine rules as needed. Suppose you wanted to find the limit of
To simplify this, we’d want to use the product law, the quotient law, and another limit law called the difference law. Our original expression becomes
So we have 3/(3-2) = 3.
Klinger-Logan, Kim. Calculus Lecture Notes, Chapter 2. Limits. Retrieved from http://www-users.math.umn.edu/~kling202/hamline/calculus/Chapter2/Limits on September 8, 2019
Feldman, Rechnitzer, & Yeager. CLP-1 Differential Calculus. 1.4.1 Calculating Limits with Limit Laws. Retrieved from http://www.math.ubc.ca/~CLP/CLP1/clp_1_dc/subsection-5.html on September 11 2019
Libretexts. Calculus Map: Calculus— Early Transcendental (Stewart). Retrieved from https://math.libretexts.org/Bookshelves/Calculus/Map%3A_Calculus_-_Early_Transcendentals_(Stewart)/2%3A_Limits_and_Derivatives/2.3%3A_Calculating_Limits_Using_the_Limit_Laws on September 8, 2019
Stephanie Glen. "Limit of Product/Quotient" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/limit-of-product-quotient/
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