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I am working on solving Limits algebraically (we haven't broken into derivatives yet or L'Hopital) and I encountered a problem that seemed very difficult to find similar type problems on the NET or in books...and I have many books...seems radicals and various odd even index types aren't that common.

Anyway here is the problem:

[tex]

\lim_{x\rightarrow 0} \frac{\sqrt[3]{1+x^2}- \sqrt[4]{1-2x}}{x+x^2}

[/tex]

2. In speaking with my teacher, she mentioned I couldn't use a conjugate because there is a mixture of odd and even radical indexes...nor could I use substitution because the terms under the radical are different so those attempts to get rid of the radical didn't work for me.

My instructor mentioned if I split the problem into a form like

[tex]

\frac{(a^3 -1) - (a^4-1)}{x+x^2}

[/tex]

a^3 -1 and a^4 -1 for the numerator...then that might be a way.

3. So my two questions are:

What does she mean by splitting the problem up like this and is there a more creative way to take care of this radical? I am very curious for creative/ new ways to approach this problem!

Thanks for any and all help.

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# Complex Indeterminate Algebraic Limit

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