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Given the function defined by the equation $$f(x)=\frac{a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}}{b_{m} x^{m}+b_{m-1} x^{m-1}+\cdots+b_{1} x+b_{0}}$$ Determine the horizontal asymptotes (if they exist) if (a) $n<m,$ (b) $n=m$, (c) $n>m$.

See Section 3.4, Theorem 2.

Algebra

Chapter 1

Functions and their Applications

Section 7

More on Functions

Functions

Campbell University

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

Lectures

01:43

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x^2. The output of a function f corresponding to an input x is denoted by f(x).

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Determine the horizontal a…

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Find the horizontal asympt…

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Determine all horizontal a…

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a. Identify the horizontal…

to find the limit of this function, we divide X to the power of end to both the numerator and the denominator and also notes that all the other terms in the numerator acts to the power of a negative and the X goes to infinity. This term out goes to zero and only the first term left, which is a constant in situation. A M is bigger and minors and is positive and enhance X to the power of a positive number goes to infinity as X goes to infinity. So the denominator goes to infinity and the numerator is a constant. The whole function goes to zero consideration be an equilibrium and hands and manners and equal zero and adds to the power of zero equals one. And I also know that in this case, all the other terms in the denominator is X to the power of a negative. And all of them go to zero as X goes to infinity so that the nominator goes to be him Enhance The limit of this function is a over B m in situations, see, any is bigger and in this case and minor sin is negative and hands X to the power of a negative number goes to zero. Now the denominator goes to zero. But the numerator is a constant and the whole limit goes to infinity as X goes into infinity. I want to mention here that the limit of this function depends on the increasing speed of the numerator and denominator. If the numerator increases faster than the denominator and limit will go to infinity and otherwise it will go to zero. Since this is a rational function, both the denominator and numerator palla nominal and the increasing speed of the phenomenal depends on the term of the highest order, which is which is determined by N m. So if n is bigger, the numerator increases faster and this limits goes to infinity. If em is bigger, the denominator goes faster and this name it goes to zero. And if I am echoes and the denominator and the numerator increases at the same speed and their name, it depends on the coefficient of the highest orders.

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