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.
