Limits at a Glance - Concept
When looking at limits, precalculus books briefly explain the concept and how to calculate basic problems. Regarding limits, Precalculus helps us understand how to calculate the value of a function at infinity. Particular types of functions, such as certain rational functions, have easier methods for calculating limits. With the introduction to limits, Precalculus gives us our first taste of Calculus.
After you solve several limits of rational functions you might start to notice patterns and I want to talk a little bit about those patterns now.
Here is a typical rational function you have f of x equals and you have a polynomial on top and a polynomial on bottom and for simplicity sake, I've just written the leading terms of each of those polynomials so we're going to assume that the numerator is a polynomial of degree n, degree n, and the denominator is a polynomial of degree m. Now we have three cases either the degree of the numerator is smaller than the degree of the denominator, it's equal or it's greater and each of those cases something different will happen with the limit of this function as x goes to infinity.
Now if the degree of the numerator is smaller you know for example the numerator is a quadratic degree 2 and the denominator is a cubic degree 3, then the limit as x approaches infinity of f of x is always going to be zero always. If the degrees are equal so say this is a cubic and this is a cubic then the limit is actually going to be the fraction of their leading coefficients, a over b and that's always true.
Now if the degree of the numerator is bigger than the degree of the denominator then one of two things can happen. It's either going to go to infinity or negative infinity and so you have to decide which which it goes to based on the Algebra extraction of the function itself.
And the other thing to remember is that these results are also identical for limit as x approaches negative infinity so it's the same results as x approaches negative infinity f of x goes to zero if the numerator has degree less than the denominator and so on.
So these are really handy really handy facts and using them you can just glance at a rational function and tell what the limit as x approaches infinity is.