Solving limit $ \lim_{x\rightarrow 1}\left(\frac{\sqrt[3]{7+x^3}-\sqrt[2]{3+x^2}}{x-1}\right) $ without L'Hopital.

Question

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$$ \lim_{x\rightarrow 1}\left(\frac{\sqrt[3]{7+x^3}-\sqrt[2]{3+x^2}}{x-1}\right) $$

Answer 1

Hint: Use the (rather lengthy, but perfectly working) identity $$a-b=\frac{a^6-b^6}{a^5+a^4b+a^3b^2+a^2b^3+ab^4+b^5}$$

Answer 2

$$\lim_{x\to1}\dfrac{(7+x^3)^{1/3}-2}{x-1}=\dfrac{d(7+y^3)^{1/3}}{dy}_{(\text{ at }y=1)}=?$$