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Limits of Rational Functions - Problem 3 2,919 views

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.

Let’s take the limit of another rational function we’ll get the limit as x approaches infinity of 2x³ minus 9x² minus 2x minus 9 over 40x² plus 8x³. Be careful here. I’ve written the bottom, the bottom polynomial with the terms out of order so when you’re using the trick of multiplying by 1 over the highest power of x in the denominator, make sure you check for the highest power of x, it’s actually x³ here.

’m going to multiply by 1 over x³ and 1 over x³ and let’s see what that gives me. I get the limit as x approaches infinity of 2x³ times 1 over x³ is 2, 9x² times 1 over x³ is 9 over x minus 2x times 1 over x² is minus 2, I’m sorry all over x³ is minus 2 over x² and I get minus 9 over x³. And then the denominator we get 40x² times 1 over x³, that’s 40 over x, and then plus 8x³ times 1 over x³, x plus 8.

What happens? Each of these guys is going to go to zero as x goes to infinity. As the denominators get big, these guys get small. And so does this guy going to zero. Now what’s left is 2 plus a bunch of zero and zero plus 8. So the final limit is 2/8 or ¼.

That’s it, always remember the trick of multiplying by the reciprocal of the highest power of x in the denominator and remember later on when you’re calculating more limits in calculus, this trick only for limit of x approaches infinity or as x approaches negative infinity.

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