<s>[INST] Find the limit of $ \lim_{x \to 7} \frac{\sqrt{x+2}-\sqrt[3]{x+20}}{\sqrt[4]{x+9}-2} $ I need to evaluate the limit without using L'Hopital's rule.
$$\Large
\lim_{x \to 7} \frac{\sqrt{x+2}-\sqrt[3]{x+20}}{\sqrt[4]{x+9}-2}
$$
 [/INST] If you are allowed to use the binomial expansion, set $y=x-7$ and examine to first order$$\frac {3\sqrt {1+\frac y9}-3\sqrt[3]{1+\frac y{27}}}{2\sqrt[4]{1+\frac y{16}}-2}=\frac {\frac 32\cdot\frac y9-\frac 33\cdot\frac y{27}}{\frac 24\cdot\frac y{16}}=\frac {112}{27}$$
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