# Video: Evaluating Limits Using Algebraic Limit Laws

Assume that lim_(𝑥 → 3) 𝑓(𝑥) = 5, lim_(𝑥 → 3) 𝑔(𝑥) = 8, and lim_(𝑥 → 3) ℎ(𝑥) = 9. Find lim_(𝑥 → 3) (𝑓(𝑥) ⋅ 𝑔(𝑥) − ℎ(𝑥)).

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### Video Transcript

Assume that the limit as 𝑥 tends to three of the function 𝑓 of 𝑥 is five, The limit as 𝑥 tends to three of the function 𝑔 of 𝑥 is eight, and the limit as 𝑥 tends to three of the function ℎ of 𝑥 is nine. Find the limit as 𝑥 tends to three of the combined function 𝑓 of 𝑥 multiplied by 𝑔 of 𝑥 minus ℎ of 𝑥.

Let 𝑢 of 𝑥 equal The combined function 𝑓 of 𝑥 multiplied by 𝑔 of 𝑥 minus ℎ of 𝑥. Let’s clarify the order of the operations in the combined function 𝑢 of 𝑥 who’s limit we are asked to find as 𝑥 approaches three. Recalling the acronym PEMDAS, we gather that multiplication comes before subtraction. So we must multiply the functions 𝑓 of 𝑥 and 𝑔 of 𝑥 together first and then subtract the function ℎ of 𝑥 in order to form the function 𝑢 of 𝑥. We want to find the limit as 𝑥 tends to three of the function 𝑢 of 𝑥. In order to do this, we will use the following properties of limits.

Number one, the limit of a difference of functions is the difference of their limits where the order in which the difference is taken is preserved. Number two, the limit of a product of functions is the product of their limits. The limit we are asked to find in the question is a difference of the combined function 𝑓 multiplied by 𝑔 and the function ℎ. So using property number one, we can rewrite the limit in question as the limit as 𝑥 tends to three of the combined function 𝑓 of 𝑥 multiplied by 𝑔 of 𝑥 minus the limit as 𝑥 tends to three of the function ℎ of 𝑥.

Next, we can use property number two to rewrite the limit as 𝑥 tends to three of the product 𝑓 of 𝑥 𝑔 of 𝑥 as the product of the limit as 𝑥 tends to three of 𝑓 of 𝑥 with the limit as 𝑥 tends to three of 𝑔 of 𝑥. Now, it just remains to substitute the limits of 𝑓, 𝑔, and ℎ as 𝑥 approaches three for the numerical values as given to us at the start of the question. The limit as 𝑥 tends to three of 𝑓 of 𝑥 equals five. The limit as 𝑥 tends to three of 𝑔 of 𝑥 equals eight. And the limit as 𝑥 tends to three of ℎ of 𝑥 equals nine. Computing five times eight minus nine, we obtain 40 minus nine, which is 31. So we obtain that the limit in question is equal to 31.