Video: Evaluating Limits Using Algebraic Limit Laws

Assume that lim_(π‘₯ β†’ 3) 𝑓(π‘₯) = 5, lim_(π‘₯ β†’ 3) 𝑔(π‘₯) = 8, and lim_(π‘₯ β†’ 3) β„Ž(π‘₯) = 9. Find lim_(π‘₯ β†’ 3) (𝑓(π‘₯) β‹… 𝑔(π‘₯) βˆ’ β„Ž(π‘₯)).

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Video Transcript

Assume that the limit as π‘₯ tends to three of the function 𝑓 of π‘₯ is five, The limit as π‘₯ tends to three of the function 𝑔 of π‘₯ is eight, and the limit as π‘₯ tends to three of the function β„Ž of π‘₯ is nine. Find the limit as π‘₯ tends to three of the combined function 𝑓 of π‘₯ multiplied by 𝑔 of π‘₯ minus β„Ž of π‘₯.

Let 𝑒 of π‘₯ equal The combined function 𝑓 of π‘₯ multiplied by 𝑔 of π‘₯ minus β„Ž of π‘₯. Let’s clarify the order of the operations in the combined function 𝑒 of π‘₯ who’s limit we are asked to find as π‘₯ approaches three. Recalling the acronym PEMDAS, we gather that multiplication comes before subtraction. So we must multiply the functions 𝑓 of π‘₯ and 𝑔 of π‘₯ together first and then subtract the function β„Ž of π‘₯ in order to form the function 𝑒 of π‘₯. We want to find the limit as π‘₯ tends to three of the function 𝑒 of π‘₯. 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 β„Ž. So using property number one, we can rewrite the limit in question as the limit as π‘₯ tends to three of the combined function 𝑓 of π‘₯ multiplied by 𝑔 of π‘₯ minus the limit as π‘₯ tends to three of the function β„Ž of π‘₯.

Next, we can use property number two to rewrite the limit as π‘₯ tends to three of the product 𝑓 of π‘₯ 𝑔 of π‘₯ as the product of the limit as π‘₯ tends to three of 𝑓 of π‘₯ with the limit as π‘₯ tends to three of 𝑔 of π‘₯. Now, it just remains to substitute the limits of 𝑓, 𝑔, and β„Ž as π‘₯ approaches three for the numerical values as given to us at the start of the question. The limit as π‘₯ tends to three of 𝑓 of π‘₯ equals five. The limit as π‘₯ tends to three of 𝑔 of π‘₯ equals eight. And the limit as π‘₯ tends to three of β„Ž of π‘₯ 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.

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