# Evaluate the following limits. lim_{(u,v)rightarrow(8,8)}frac{u^{1/3}-v^{1/3}}{u^{2/3}-v^{2/3}}

Evaluate the following limits.
$\underset{\left(u,v\right)\to \left(8,8\right)}{lim}\frac{{u}^{1/3}-{v}^{1/3}}{{u}^{2/3}-{v}^{2/3}}$
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insonsipthinye
Consider the provided expression,
$\underset{\left(u,v\right)\to \left(8,8\right)}{lim}\frac{{u}^{1/3}-{v}^{1/3}}{{u}^{2/3}-{v}^{2/3}}$
Evaluate the following limits.
Multiplying by the conjugate of the numerator and simplifying further,
$\underset{\left(u,v\right)\to \left(8,8\right)}{lim}\frac{{u}^{1/3}-{v}^{1/3}}{{u}^{2/3}-{v}^{2/3}}=\underset{\left(u,v\right)\to \left(8,8\right)}{lim}\left(\frac{\frac{{u}^{\frac{2}{3}}-{v}^{\frac{2}{3}}}{{u}^{\frac{1}{3}}+{v}^{\frac{1}{3}}}}{{u}^{\frac{2}{3}}-{v}^{\frac{2}{3}}}\right)$
$=\underset{\left(u,v\right)\to \left(8,8\right)}{lim}\left(\frac{1}{{u}^{\frac{1}{3}}+{v}^{\frac{1}{3}}}\right)$
$=\frac{1}{{8}^{\frac{1}{3}}+{8}^{\frac{1}{3}}}$
$=\frac{1}{4}$
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