# Don't understand how to simplify this limit

• zeion
In summary, the conversation discusses the process of finding the limit of a function involving fractions and reciprocal multiplication. The expert suggests using more parentheses for clarity and recommends changing variables to make the problem easier to solve. The conversation also mentions factoring and using polynomial division to simplify the expression.
zeion

## Homework Statement

lim
x->-1 x^1/3 + 1 / x + 1 = x^1/3 + 1 / x + 1 ((x^2/3 - x^1/3 + 1) / (x^2/3 - x^1/3 + 1))

= x + 1 / (x + 1)(x^2/3 - x^1/3 + 1)

cancel out and done

I don't understand how to know what reciprocal to multiply in cases like these to make it work.

Your formatting is horrible. Use more parentheses, ok? I think you mean limit x->(-1) of (x^(1/3)+1)/(x+1). It might be a little clearer if you change variables first and let u=x^(1/3). So u->(-1) also and now your expression is (u+1)/(u^3+1). Can you factor (u^3+1)=(u+1)*(something)? Use polynomial division to divide u^3+1 by u+1 if you don't know the answer.

It seems like you are trying to simplify the given limit by manipulating the expression algebraically. This can be a helpful approach, but it is important to remember that there are certain rules and properties that need to be followed in order to make the simplification accurate. In this case, it looks like you are trying to use the rule for simplifying fractions, where you multiply the numerator and denominator by the same reciprocal. However, in this expression, the numerator and denominator are not simple fractions, but rather expressions with exponents. Therefore, you need to use the appropriate rules for simplifying expressions with exponents. In this case, you can use the rule for adding and subtracting exponents to simplify the numerator and denominator separately before applying the fraction simplification rule. I suggest reviewing the rules for simplifying expressions with exponents and practicing more examples to better understand how to approach problems like this.

## What does it mean to "simplify" a limit?

Simplifying a limit means to find a value that the function approaches as the input of the function approaches a specific value. This value is known as the limit.

## How do I know if a limit can be simplified?

Limits can be simplified if the function is continuous at the specific input value. This means that the function has a defined value at that point and no gaps or jumps in the graph.

## What is the process for simplifying a limit?

To simplify a limit, you can try substituting the specific input value into the function and see if it produces a defined value. If it does, then that value is the limit. If it does not, you may need to use algebraic techniques such as factoring or rationalizing the denominator to find the limit.

## What should I do if I am still having trouble simplifying a limit?

If you are having trouble simplifying a limit, it may be helpful to graph the function to see if there are any gaps or jumps at the specific input value. Also, consulting with a teacher or tutor can provide additional guidance and support.

## What are some common mistakes to avoid when simplifying a limit?

Some common mistakes to avoid when simplifying a limit include forgetting to check for continuity, making arithmetic errors while substituting values, and not simplifying the resulting expression fully. It is also important to understand the properties of limits, such as the sum and product rules, to avoid making mistakes while simplifying.

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