# Finding the Limit of a Rational Function

• Manarius
In summary, the conversation is about finding the limit of a function as x approaches 0. The function is (1/(x(x+1)^1/2)) - (1/x) and the problem is an algebra problem rather than a calculus problem. The person is struggling to remember how to get the limit of the denominator to not equal 0 and has tried various methods without success. They ask for clarification on the solution suggested by the other person, which involves factoring and multiplying and dividing by a different expression.
Manarius
This is just a problem I came across while reviewing basic calculus.

## Homework Statement

Find the limit as x approaches 0 of f(x)=(1/(x(x+1)^1/2)) - (1/x)

## The Attempt at a Solution

My problem here is really more of an algebra problem than a calculus problem. I cannot for the life of me remember how to get the limit of the denominator of either term to not equal 0. I've tried everything I can think of (rationalizing the denominator, etc.), to no avail.

I could find no tutorials either online or in my book that use an example quite like this one. It's truly maddening, especially because I know I should have learned this 6 years ago in Algebra.

Thanks in advance. Hope you can help.

Do you mean

$$f(x)=\frac{1}{x\sqrt{x+1}}-\frac{1}{x}$$

?

If yes, factor out 1/x and multiply and divide f(x) by [1/sqrt(x+1)+1].

ehild

I think I must be misunderstanding what you're saying. Either that or I did the math wrong multiple times.

Would you mind clarifying what exactly you mean? Thanks.

Use the notation 1/√(x+1)=a.

f(x)=(1/x) (a-1)

Multiply and divide by (a+1)

f(x)=(1/x)[(a-1)(a+1)]/(a+1)

f(x)=(1/x)(a2-1)/(a+1).

Replace back 1/√(x+1) for a and simplify. Find the limit.

## 1. What is the definition of a rational function?

A rational function is a mathematical expression that can be written as a ratio of two polynomials, where the denominator is not equal to zero.

## 2. What is the limit of a rational function?

The limit of a rational function is the value that the function approaches as the input or independent variable approaches a certain value.

## 3. How do you find the limit of a rational function?

To find the limit of a rational function, you can substitute the value that the independent variable is approaching into the function and simplify the resulting expression. If the simplified expression has a finite value, that value is the limit of the function.

## 4. What are some common approaches for evaluating the limit of a rational function?

Some common approaches for evaluating the limit of a rational function include direct substitution, factoring the numerator and denominator, and using algebraic manipulation or the rules of limits.

## 5. What is the significance of finding the limit of a rational function?

Finding the limit of a rational function is important in many applications, such as in calculus, physics, and engineering. It allows us to understand the behavior of a function near a certain point and can help us make predictions and solve problems in these fields.

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