<s>[INST] How do I know when the limit of a function at a certain point doesn't exist? $$\lim_{x \to 8} \frac{\sqrt{7+\sqrt[3]{x}}-3}{x-8}$$
I have this limit. I can't use L'Hopital. After rationalizing the numerator I get:
$$\lim_{x \to 8} \frac{\sqrt[3]{x}-2}{(x-8)(\sqrt{7+\sqrt[3]{x}}+3)}$$
Isn't there anything left to do after that? How to know if the limit just doesn't exist?
EDIT: I typed the limit wrong.
 [/INST] \begin{align}
\lim_{x \to 8} \frac{\sqrt{7+\sqrt[3]{x}}-3}{x-8}
&=\lim_{x \to 8} \frac{\sqrt{7+\sqrt[3]{x}}-3}{x-8} \cdot \frac{\sqrt{7+\sqrt[3]{x}}+3}{\sqrt{7+\sqrt[3]{x}}+3}\tag{1}\\
&=\lim_{x \to 8} \frac{(7+\sqrt[3]{x})-9}{(x-8)(7+\sqrt[3]{x}+3)}\tag{2}\\
&=\lim_{x \to 8} \frac{\sqrt[3]{x}-2}{(x-8)(7+\sqrt[3]{x}+3)}\tag{3}\\
&=\lim_{x \to 8} \frac{\sqrt[3]{x}-2}{(\sqrt[3]{x}^3-2^3)(7+\sqrt[3]{x}+3)}\tag{4}\\
&=\lim_{x \to 8} \frac{\sqrt[3]{x}-2}{(\sqrt[3]{x}-2)(\sqrt[3]{x}^2+2\sqrt[3]{x}+4)(7+\sqrt[3]{x}+3)}\tag{5}\\
&=\lim_{x \to 8} \frac{1}{(\sqrt[3]{x}^2+2\sqrt[3]{x}+4)(7+\sqrt[3]{x}+3)}\tag{6}\\[6pt]
&=\frac{1}{(4+4+4)(7+4+3)}=\frac{1}{168}\tag{7}
\end{align}
Explanation:  
$(2)$: I used $(x-y)(x+y)=(x^2-y^2)$
$(5)$: I used $(x^3-y^3)=(x-y)(x^2+xy+y^2)$  
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