Don't understand how to simplify this limit

[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.
Please help.

 

[Dick]

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.

 

[Architeuthis Dux]

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.