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This is problem number 45 of stewart calculus 8th edition, section 2 .2.
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Determine the limit as x approaches 1 from the left and as x approaches 1 from the right of the function, 1 divided by the quantity x cubed minus 1.
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For part a, we're going to evaluate these limits for this given function, for values of x that approach 1 from the left and from the right.
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So if we start right there, we're going to choose values.
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We're going to make a table of values of numbers that are very close to one, both from the left and from the right.
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Beginning with from the right, here we see that as we choose values closer and closer to positive 1 from the right, we see the value of the function 1 over x cubed minus 1.
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We see this value increasing without bound.
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And as we approach x from the right, as we approach 1 from the right for x, we see that the value of the function approaches positive infinity.
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And this is the behavior we see for the second limit as x approaches 1 from the right.
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If we take a look at the other values down here, these are values that are chosen close to the value of 1.
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However, smaller than one or less than one.
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So these are approaching one from the left.
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And as we get closer and closer to one from the left, you see that the values decrease or that bound.
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They are increasingly negative, and this we interpret as approaching negative infinity for this first limit.
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In part b, we're going to try to determine these exact same results for these limits.
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By using the reasonings from example 9 in the textbook...