# Find Limit of f(x) w/o L'Hopitals Rule

In summary, the conversation is about finding the limit of a function as x approaches infinity and whether L'hopital's rule can be used. The conclusion is that L'hopital's rule is not needed and the limit can be found by simplifying the function.

## Homework Statement

Find the limit of f(x)=[xcosx]/[x^3 + 1] as x tends to infinity.

## The Attempt at a Solution

Can i use l'hopitals rule here? Or if not, what are the conditions for f(x) to meet so that i may use l'hopitals rule?

Without using l'hop i know that cosx would oscillate between -1 and 1 and that is multiplied by infinity. Then the denominator would tend towards infinity. So what is infinity over infinity?

I think that you don't need L'hopital here.
i.e, if we write it as:
(cos(x)/x^2)*1/(1+1/x^3)->0
as x appraoches infinity, can you see why?

yea, thank you!

## 1. What is the definition of a limit?

A limit is the value that a function approaches as the input (x) approaches a specific value. It is denoted by the notation lim f(x) as x approaches a. Essentially, it represents the behavior of a function near a particular point.

## 2. How do you find the limit of a function without using L'Hopital's Rule?

There are several methods for finding the limit of a function without using L'Hopital's Rule, such as direct substitution, factoring, and rationalizing. You can also use algebraic techniques, such as taking the conjugate or finding the common denominator, to simplify the function and evaluate the limit.

## 3. What is the purpose of finding the limit of a function?

The limit of a function helps us understand the behavior of the function near a specific point. It can also be used to determine if a function is continuous at a particular point and to evaluate indeterminate forms.

## 4. Can you find the limit of a function at a point where it is not defined?

No, the limit of a function can only be found at points where the function is defined. If the function is not defined at a particular point, the limit does not exist at that point.

## 5. What are some common mistakes when finding the limit of a function without using L'Hopital's Rule?

Some common mistakes when finding the limit of a function without using L'Hopital's Rule include forgetting to simplify the function, dividing by zero, and not considering the behavior of the function at the point in question. It is important to carefully evaluate the function and use proper algebraic techniques to avoid these errors.

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