Unit
Polynomial and Rational Functions
Cornell University
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
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Cornell University
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
We're talking about limits of rational functions and we have some tricks that allow us to find these limits at a glance, let's look at an example.
A limit is x approaches infinity of -3x² plus 12x over 2x² minus 7x minus 30. Now because the degree of the numerator and the degree of the denominator of the same, we can look just at the leading coefficients are the numerator and the denominator, so we have a -3 here and a 2 here and the limits is going to be the fraction of those coefficients -3 over 2 that's it that's our answer.
Now it's trickier when the numerator has a higher degree than the denominator, the limit is either going to be plus or minus infinity and so you have to figure out it's going to be same thing for x approaching a negative infinity. Now these are same expressions in both of these limits so let's think about both of these problems at once.
Here as x goes to positive infinity, think about what the leading term does. As x gets really large in magnitude, it's the term with the highest power of x that matters the most. So this -x³ is going to dominate as x gets large, so we're going to have a negative value when x gets really, really large in the numerator. But in the denominator this term dominates and as x gets large, we'll have a positive value down here, so big negative over big positive we're going to get negative infinity here.
As x goes to negative infinity, the numerator as x gets very large in the negative direction, we're going to get a negative value for x³ and the so the minus in front is going to make that positive, we'll get a positive value in the numerator and as x goes to negative infinity this quadratic is going to be positive no matter what right? The negative say -1000 you square that and you're going to get a positive one million, so this is going to get positive and the numerator is going to get positive, positive over positive, this means it's going to infinity.
It's got to go to either plus or minus infinity so it's just a question of which and so when you're analyzing rational functions, look at the leading term. The leading term is really what's going to determine what happens to both the numerator function and the denominator.
So again the numerator here is going to negative infinity, the denominator is going to positive, so this thing is going to negative infinity. Here the numerator is going to positive infinity and the denominator is positive, so positive infinity.