$\displaystyle\lim_{x \to 2} \dfrac{\sqrt{9-x^3}-\sqrt[3]{x^2-3}}{x^2-4}\\ =\displaystyle\lim_{x \to 2} \dfrac{\sqrt{9-x^3}-1-\sqrt[3]{x^2-3}+1}{x^2-4}\\ =\displaystyle\lim_{x \to 2} \dfrac{\sqrt{9-x^3}-1}{x^2-4}-\displaystyle\lim_{x \to 2} \dfrac{\sqrt[3]{x^2-3}-1}{x^2-4}\\ =\displaystyle\lim_{x \to 2} \dfrac{(\sqrt{9-x^3}-1)(\sqrt{9-x^3}+1)}{(x-2)(x+2)(\sqrt{9-x^3}+1)}-\displaystyle\lim_{x \to 2} \dfrac{(\sqrt[3]{x^2-3}-1)\left(\sqrt[3]{x^2-3}^2+\sqrt[3]{x^2-3}+1\right)}{(x-2)(x+2)\left(\sqrt[3]{x^2-3}^2+\sqrt[3]{x^2-3}+1\right)}\\=\displaystyle\lim_{x \to 2} \dfrac{8-x^3}{(x-2)(x+2)(\sqrt{9-x^3}+1)}-\displaystyle\lim_{x \to 2} \dfrac{x^2-4}{(x^2-4)\left(\sqrt[3]{x^2-3}^2+\sqrt[3]{x^2-3}+1\right)}\\=\displaystyle\lim_{x \to 2} \dfrac{(2-x)(4+2x+x^2}{(x-2)(x+2)(\sqrt{9-x^3}+1)}-\displaystyle\lim_{x \to 2} \dfrac{1}{\left(\sqrt[3]{x^2-3}^2+\sqrt[3]{x^2-3}+1\right)}\\=-\displaystyle\lim_{x \to 2} \dfrac{4+2x+x^2}{(x+2)(\sqrt{9-x^3}+1)}-\displaystyle\lim_{x \to 2} \dfrac{1}{\left(\sqrt[3]{x^2-3}^2+\sqrt[3]{x^2-3}+1\right)}\\=- \dfrac{4+2.2+2^2}{(2+2)(\sqrt{9-2^3}+1)}- \dfrac{1}{\left(\sqrt[3]{2^2-3}^2+\sqrt[3]{2^2-3}+1\right)}\\=-\dfrac{11}{6}$
