WEBVTT
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Given that the limit as π₯ approaches three of π of π₯ over four π₯ squared is equal to negative four, determine the limit as π₯ approaches three of π of π₯ over π₯.
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To establish this limit, weβre going to recall a few of our limit laws.
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The first law we might recall is that the limit of the product of two functions is equal to the product of the limit of those two functions.
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So, what weβre going to begin by doing is writing π of π₯ over four π₯ squared as the product of two functions, as the product of π of π₯ over π₯ and one over four π₯.
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And the reason we did this is we now have a function that looks a little bit like the limit weβre trying to find.
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And we can now split this limit up as the limit as π₯ approaches three of π of π₯ over π₯ times the limit as π₯ approaches three of one over four π₯.
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Weβre now going to apply direct substitution to actually evaluate the limit as π₯ approaches three of one over four π₯.
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Itβs one over four times three, which is one twelfth.
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Weβre going to put this constant in front of our other limit and we see that our limit becomes a twelfth times the limit as π₯ approaches three of π of π₯ over π₯.
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And of course, this is still our original limit so itβs equal to negative four.
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We can now solve this limit equation by multiplying both sides by 12.
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12 times negative four is negative 48.
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And we see that we actually have the solution to this question.
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The limit as π₯ approaches three of π of π₯ over π₯ is equal to negative 48.