How can we evaluate this limit without using l'Hôpital's rule?

[Jameson]

Evaluate the following limit without using l'Hôpital's rule.

$$\lim_{x\to 1}\frac{1}{2(1 - \sqrt{x})} - \frac{1}{3(1 - \sqrt[3]{x})}$$.

Hint:

Spoiler

Use the substitution $x=u^6$.

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[Jameson]

Congratulations to the following members for their correct solutions:

1) MarkFL
2) anemone
3) Sudharaka
4) soroban

Solution (from Sudharaka):

Spoiler

Substitute \(x=u^6\) and we get,

\begin{eqnarray}

\lim_{x\to 1}\left(\frac{1}{2(1 - \sqrt{x})} - \frac{1}{3(1 - \sqrt[3]{x})}\right)&=&\lim_{u\to 1}\left(\frac{1}{2(1 - u^3)} - \frac{1}{3(1 - u^2)}\right)\\

&=&\frac{1}{6}\lim_{u\to 1}\left[\frac{1-3u^2+2u^3}{(1 - u^2)(1-u^3)}\right]\\

&=&\frac{1}{6}\lim_{u\to 1}\left[\frac{(1-u)^2 (1+2u)}{(1 - u^2)(1-u^3)}\right]\\

&=&\frac{1}{6}\lim_{u\to 1}\left[\frac{1+2u}{(1+u)(1+u+u^2)}\right]\\

&=&\frac{1}{12}

\end{eqnarray}