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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.
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Find the limit as π₯ tends to three of the combined function π of π₯ multiplied by π of π₯ minus β of π₯.
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Let π’ of π₯ equal The combined function π of π₯ multiplied by π of π₯ minus β of π₯.
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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.
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Recalling the acronym PEMDAS, we gather that multiplication comes before subtraction.
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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₯.
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We want to find the limit as π₯ tends to three of the function π’ of π₯.
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In order to do this, we will use the following properties of limits.
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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.
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Number two, the limit of a product of functions is the product of their limits.
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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.
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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₯.
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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₯.
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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.
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The limit as π₯ tends to three of π of π₯ equals five.
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The limit as π₯ tends to three of π of π₯ equals eight.
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And the limit as π₯ tends to three of β of π₯ equals nine.
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Computing five times eight minus nine, we obtain 40 minus nine, which is 31.
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So we obtain that the limit in question is equal to 31.