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Limits of Rational Functions - Problem 2

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

I’m finding the limit of another rational function. This time I’m finding the limit as x approaches negative infinity. You shall that the method’s exactly the same as when the limit as x approaches infinity. The first thing I want to do is rewrite the limit.

I want to take this expression and I’m going to multiply the top and bottom by the highest power of x in the denominator. So I have 3x² minus 13x plus 4 over x² minus 9. I multiplied by the highest power of x in the denominator and that’s x². 1 over x², 1 over x².

Now what’s this going to give me? In the numerator I’m going to get 3x² times 1 over x² to 3. And here I’ll get minus 13 x times 1 over x² is minus 13 over x. and here I get plus 4 over x². In the denominator I get x² times 1 over x², 1 minus 9 over x².

Now you’ll notice that what this method does is it makes it so that every one of these terms either approaches a finite or it goes to zero as x goes to either infinity or negative infinity. It doesn’t actually matter. X is going to negative infinity but still 13 over x is going to zero. 4 over x² is going to zero. The denominators are getting bigger and bigger, 9 over x² is going to zero. What we’re left with is 3 minus zero plus zero over 1 minus zero, 3.

That’s it, don’t forget the trick of multiplying by the reciprocal of the highest power of x in the denominator.

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