<s>[INST] Evaluating $\lim_{x \to 1} \frac{1 - x^{1/\pi}}{1 - x^{1/e}}$ without de l'Hospital rule Is it possible to evaluate this limit without de l'Hopital's rule?
$$\lim_{x \to 1} \frac{1 - x^{1/\pi}}{1 - x^{1/e}}$$
 [/INST] Why not to use
$$1-x^a=-a (x-1)+\frac{1}{2} \left(a-a^2\right) (x-1)^2+O\left((x-1)^3\right)$$
$$\frac {1-x^a}{1-x^b}=\frac{-a (x-1)+\frac{1}{2} \left(a-a^2\right) (x-1)^2+O\left((x-1)^3\right)}{-b (x-1)+\frac{1}{2} \left(b-b^2\right) (x-1)^2+O\left((x-1)^3\right)}$$
$$\frac {1-x^a}{1-x^b}=\frac{a}{b}+\frac{a  (a-b)}{2 b}(x-1)+O\left((x-1)^2\right)$$ which shows the limit and also how it is approached.
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