# Find Infinite limit (Rational Function)underset (x->5^+)(lim)(1/x^(4/3)-1/(x-5)^(4/3))

Find Infinite limit (Rational Function)
$\underset{x\to 5}{lim}\left(\frac{1}{{x}^{4/3}}-\frac{1}{\left(x-5{\right)}^{4/3}}\right)$

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Cristiano Sears

Given:
$\underset{x\to {5}^{+}}{lim}\left(\frac{1}{{x}^{4/3}}-\frac{1}{\left(x-5{\right)}^{4/3}}\right)$
It is known that
$\underset{x\to a}{lim}\left[f\left(x\right)+-g\left(x\right)\right]=\underset{x\to a}{lim}f\left(x\right)+-\underset{x\to a}{lim}g\left(x\right)$
Then above can be written as
$=\underset{x\to {5}^{+}}{lim}\left(\frac{1}{{x}^{4/3}}\right)-\underset{x\to {5}^{+}}{lim}\frac{1}{\left(x-5{\right)}^{4/3}}$
Plug Limits
$\left(\frac{1}{{5}^{4/3}}\right)-\left(\mathrm{\infty }\right)\left(since,\left(x-5{\right)}^{4/3}>0\right)$
$-\mathrm{\infty }\left(since,c-\mathrm{\infty }=-\mathrm{\infty }\right)$
Hence,
$\underset{x\to {5}^{+}}{lim}\left(\frac{1}{{x}^{4/3}}-\frac{1}{\left(x-5{\right)}^{4/3}}\right)=-\mathrm{\infty }$