# Compute a rational limit

I tried to calculate this limit using change of variable: $\displaystyle\lim_{x\to1}\frac{\sqrt[2]{2-x} - 1}{1 + \sqrt[5]{x - 2}}$ But i don't get the result, which is -5/2. I would appreciate if somebody can help me. Thanks.

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Hint: If you substitute $2-x = y^{10}$ with $y > 0$, then $\sqrt{2-x} = y^5$ and $\sqrt[5]{x-2} = -y^2$.

Your limit then becomes $\displaystyle\lim_{y \to 1}\dfrac{y^5-1}{1-y^2}$. Can you finish from here?

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Sure thank you!, but how do you know what appropriate change of variable it will be useful? Any technique? –  egarro Aug 22 at 5:46
I don't like square roots or 5th roots very much, so I picked a substitution that got rid of them. By setting $2-x = y^{10}$, both the square root and the $5$-th root disappeared. –  JimmyK4542 Aug 22 at 5:50
That's great! Thanks a lot. –  egarro Aug 22 at 5:57