ShmoopTube

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Here's your shmoop du jour: Evaluate the limit as x approaches 11 from

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the negative side of x plus 11 over x squared minus 121.

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Here are the potential answers...

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Whenever we're given a limit problem like this and asked to evaluate it, the first thing

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we can do is just try plugging in 11 directly.

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On the top of the fraction, 11 plus 11 works out as 22...

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...but when we plug it in to the denominator, we get that 11 squared is 121

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and 121 minus 121 is 0.

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Dividing by zero is a big no-no. You can't do that!

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So to avoid that situation, let's try simplifying the bottom.

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x squared minus 121 looks like a difference of two squares...

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...a squared minus b squared equals a plus b times a minus b.

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The square root of 121 is 11. So applying the formula, the bottom simplifies to x plus

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11 times x minus 11.

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The x plus 11 on the top and bottom of the fraction cancel out... and we're left with the

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limit as x approaches 11 from the negative side of 1 over x minus 11.

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Back to the question...it asks us to take the limit as x approaches 11 from the negative side.

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Whenever we deal with limits, it means we get really close to a number from the negative

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side but never actually kiss it.

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If we try plugging in 11 as x again...11 minus 11 still gives us 0 in the denominator...

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...so we can immediately eliminate (A), (B), and (C) because this function is not defined.

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So we're left with D) or (E). We can now start plugging in values smaller

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than 11 that get closer and closer to 11...since we're approaching 11 from the negative side.

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If x equals 10, we get 1 over 10 minus 11, or negative 1.

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If x equals 10.5... then 1 over negative .5 gives us

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an even larger negative number...or negative 2.

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Looks like we're getting more and more negative....and will keep approaching negative infinity...

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So our answer is (E). Done!

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