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Any help in computing the limit of the function $|x|^{(1/|y|)}$ at the origin will be appreciated.

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    $\begingroup$ When $0<y<1$ we have $1/y>1$ So if $0<x<1$ we have $x^{1/y} < x$. Is that enough? Also: is $|x|^{1/|y|}$ defined on the $x$-axis? $\endgroup$ – GEdgar Mar 29 at 20:36
  • $\begingroup$ It's OK, Dear GEdgar $\endgroup$ – M.R. Yegan Mar 29 at 20:42
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Assume $x>0$, $y>0$, then $$ x^{1/y} = e^{(\ln x)\cdot(1/y)}. $$ As $x\to 0$ and $y\to 0$, we have $$ \ln x\to-\infty, \quad 1/y\to + \infty, $$ therefore, $$ (\ln x)\cdot(1/y)\to-\infty, $$ which implies that $$ x^{1/y} = e^{(\ln x)\cdot(1/y)} \to 0. $$

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