### Video Transcript

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

Let π’ of π₯ equal The combined
function π of π₯ multiplied by π of π₯ minus β of π₯. Letβs clarify the order of the
operations in the combined function π’ of π₯ whoβs 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
π of π₯ and π of π₯ together first and then subtract the function β of π₯ in order
to form the function π’ of π₯. We want to find the limit as π₯
tends to three of the function π’ 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 π multiplied by π and the
function β. So using property number one, we
can rewrite the limit in question as the limit as π₯ tends to three of the combined
function π of π₯ multiplied by π of π₯ minus the limit as π₯ tends to three of the
function β of π₯.

Next, we can use property number
two to rewrite the limit as π₯ tends to three of the product π of π₯ π of π₯ as
the product of the limit as π₯ tends to three of π of π₯ with the limit as π₯ tends
to three of π of π₯. Now, it just remains to substitute
the limits of π, π, and β 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
π of π₯ equals five. The limit as π₯ tends to three of
π of π₯ equals eight. And the limit as π₯ tends to three
of β 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.