# Exploring the Limits of f(x): P(x)/Q(x)

• EV33
In summary, the problem asks to find the different values of f(x), defined as the limit of P(n)/Q(n) as n approaches infinity. Using the given information, including the definition of a limit and the squeeze principle, we can prove that when p<q, f(x) is equal to 0; when q<p, f(x) is equal to infinity; and when p=q, f(x) is equal to the ratio of the leading coefficients a0/b0. The question also addresses whether it is legal to divide the top and bottom of the fraction by n^p and n^q, respectively, and concludes that it is not permissible as the resulting ratio is not equal to 1.
EV33

## Homework Statement

P(x)=a0*x^(p)+a1*x^(p-1)+a2*x^(p-2)...
Q(x)=b0*x^(q)+b1*x^(q-1)+b2*x^(q-2)...

f(x)=lim (n$$\rightarrow$$$$\infty$$)P(n)/Q(n)

prove what f(x) is equal to for when p<q,q<p,p=q

Find the different values of f(x)

## Homework Equations

The only information that I know I can use is the definition of a limit, the squeeze principle,, 1/n converges as n goes to infinity, the limit of a^n where a is a fixed number, and that you can multiply, add, and divide limits.

## The Attempt at a Solution

Let
P(n)/Q(n)=(a0*n^(p)+a1*n^(p-1)+a2*n^(p-2)...)/(b0*n^(q)+b1*n^(q-1)+b2*n^(q-2)...)
I'll just start off by asking my first question. It is not legal to divide the top by n^p and the bottom by n^q correct? I would assume its not ok to do because the ratio between the two is not 1.

Last edited:
If my wording isn't clear please let me know and I can reword what I said.

## 1. What is the purpose of exploring the limits of f(x)?

The purpose of exploring the limits of f(x) is to understand the behavior of a function as it approaches a certain value or as the input values become infinitely large or small. This can help us to better understand the behavior and properties of a function and its graph.

## 2. How do you calculate the limit of a function?

To calculate the limit of a function, we can use algebraic manipulation, substitution, or graphical methods. We can also use the formal definition of a limit, which involves evaluating the function at values closer and closer to the limit value and observing the trend of the outputs.

## 3. What are the types of limits that can be explored?

The two main types of limits that can be explored are one-sided limits and two-sided limits. One-sided limits involve approaching the limit from either the left or right side of the function, while two-sided limits consider both sides simultaneously.

## 4. How do you determine if a limit exists?

A limit exists if the value of the function approaches a finite value as the input values approach a certain value. This means that the left and right-sided limits must be equal to each other. If the left and right-sided limits are not equal, then the limit does not exist.

## 5. What are the applications of exploring the limits of f(x)?

The exploration of limits has various applications in mathematics, physics, and engineering. It can be used to model physical phenomena, analyze the behavior of systems, and solve optimization problems. Limits are also important in calculus, as they are used to define derivatives and integrals.

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