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 Mar 11 accepted Prove whether series converges or not? Mar 10 awarded Nice Question Mar 9 asked Prove whether series converges or not? Feb 29 awarded Popular Question Feb 17 comment Prove continuity for cubic root using epsilon-delta @Ishfaaq Just found your nice effort here as I am working on showing $f(x)=x^{1/3}$ is continuous at any real number $a$. However, I think there might be an error here. I agree with your effort to get $$\frac{1}{2|a|^2}<\frac{1}{|ax|}$$, which means that $$\frac{1}{2^{1/3}|a|^{2/3}}<\frac{1}{|ax|^{1/3}},$$ but note that $$\frac{|x-a|}{|ax|^{1/3}}=|x-a|\cdot\frac{1}{|ax|^{1/3}}>|x-a|\cdot\frac{1}{2^{‌​1/3}|a|^{2/3}}.$$ Jan 18 comment How does Mathematica calculate sin(Pi/5)? I had to do FunctionExpand[Sin[Pi/8]] to get the answer. Jan 16 comment How does Mathematica calculate sin(Pi/5)? @Lubin Yes I do, thank to Simon Woods link. Jan 16 asked How does Mathematica calculate sin(Pi/5)? Jan 11 revised Limit of $\sin(x)$ as $x$ approaches zero from the left added 427 characters in body Jan 5 awarded Yearling Dec 27 revised Prove that the $\lim_{x\to a}\sqrt[n]{x}=\sqrt[n]{a}$ added 137 characters in body; edited title Dec 27 comment Prove that the $\lim_{x\to a}\sqrt[n]{x}=\sqrt[n]{a}$ I'd love to see the detailed steps showing a proof of this inequality. Dec 27 comment Prove that the $\lim_{x\to a}\sqrt[n]{x}=\sqrt[n]{a}$ Also, it seems that if $a>0$, then $x^{1-1/n}+a^{1/n}x^{1-2/n}+a^{2/n}x^{1-3/n}+\cdots+a^{1-1/n}$ is simply greater than $(a/2)^{1-1/n}$ if $x>a/2$. So you don't have to deal with the terms after $x^{1-1/n}$, but it was very, very cool the way you dealt with them. Dec 27 accepted Prove that the $\lim_{x\to a}\sqrt[n]{x}=\sqrt[n]{a}$ Dec 27 comment Prove that the $\lim_{x\to a}\sqrt[n]{x}=\sqrt[n]{a}$ Very nice answer. This is right at the level I need when Calc I students are seeing this limit for the first time and are still in the second caper. Had to do a little work to verify $\ge n(a/2)^{1-1/n}$. Now, what about if $a<0$? How about a specific example, say $\lim_{x\to-5}\sqrt[3]{x}=\sqrt[3]{-5}$? Dec 27 asked Prove that the $\lim_{x\to a}\sqrt[n]{x}=\sqrt[n]{a}$ Dec 27 revised Limit of $\sin(x)$ as $x$ approaches zero from the left added 336 characters in body Dec 27 comment Limit of $\sin(x)$ as $x$ approaches zero from the left Thanks for these ideas. Very useful. Dec 27 comment Limit of $\sin(x)$ as $x$ approaches zero from the left Thanks for the help. Dec 26 comment Limit of $\sin(x)$ as $x$ approaches zero from the left @ThePortakal You can use SOHCAHTOA or the unit circle. See my update.