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.

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Limits at a Glance - Problem 2

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.

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We're talking about limits of rational functions and we found some tricks that allow us to find the limits at a glance. For example the limit is x approaches negative infinity of 2 over x³ plus 4. Just because of the degree of the denominator is bigger than the degree of the numerator, this automatically goes to 0, as x goes to negative infinity. It would be the same if x were going to positive infinity.

Now here as I go to negative infinity, what's the limit is x goes to negative infinity of 2x³ over x³ plus 4? Because the degrees are the same they both have degree 3, we look at the leading coefficients, we get a 2 in the numerator and a marginal 1 in front of that x³. This limit will be 2 over 1 or 2 and it would be the same limit if x were approaching positive infinity.

Now here is where things gets a little trickier, the limit is x approaches negative infinity of 2x to the fourth over x³ plus 4, in this case the numerator has higher degree than the denominator so it's either going to be plus or minus infinity, so which is it? Well as x goes to the infinity, you have to remember that x is going to be negative, well the numerators is always going to be positive because you're raising to the fourth power, you raise any number positive or negative to the fourth power, you will get a positive number, but x³ will be negative and when it gets big enough, the whole denominator will be negative.

So the denominator will be negative when x is way, way down there say -1000, negative a million this will be negative, this will be positive it's got to go to negative infinity and it maybe different as x goes to positive infinity, it may go to positive infinity, so you got to make sure you check the signs of the numerator and denominator to be sure whether it goes to positive or negative infinity.