calculate limit without L'Hospital's Rules

Question

I'm struggling with this limit. We did not learn L'Hospital's Rule, Taylor expansions, etc. yet. Should be solvable with plain old limit arithmetic:

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

I tried combining the fractions with the common denominator, I tried multiplying each one by the conjugates, substituting different values as t.

Still haven't had my breakthrough yet.

Please help, thanks

Answer 1

Hint: We can substitute $x=t^6$ to obtain $$ \lim_{t\to 1}\frac{3}{1-t^3}-\frac{2}{1-t^2}=\lim_{t\to 1}\frac{1}{1-t}\left(\frac{3}{t^2+t+1}-\frac{2}{1+t}\right). $$