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 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. 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 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 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. Dec 26 comment Limit of $\sin(x)$ as $x$ approaches zero from the left @angryavian I am trying to prove the continuity. See my update. Oct 15 comment Multivariable Chain Rule for partially differentiable maps @user251257 I have a reply to this answer posted as another answer. Love to hear if I am doing it correctly. Aug 2 comment Critical value example where partial derivative does not exist @RecklessReckoner Nice job. Thanks for the help. It will be very useful teaching class. Aug 1 comment Critical value example where partial derivative does not exist @RecklessReckoner How about an example where you get two critical values, one where both partial derivatives equal zero and one where at least one of the partial derivatives is undefined? Jun 20 comment Existence of kth moment I finally figured out your answer, another great answer, but you are right, a huge overestimate. Here is some Mathematica code comparing the actual integral and your bound for $k=6$. In[3]:= k = 6 Out[3]= 6 In[7]:= Integrate[ Abs[x]^k*Exp[-(x - 3)^2], {x, -[Infinity], [Infinity]}] // N Out[7]= 2551.67 In[8]:= Integrate[Abs[x + 3]^k, {x, -1, 1}] + 2^(2*k + 1)*(k + 1)^k // N Out[8]= 9.63783*10^8 Jun 19 comment Existence of kth moment $|-2|>1$, but $|-2+3|$ is not less than $4(-2)$. Jun 19 comment Existence of kth moment And that's only to the right of $x=1$. Jun 19 comment Existence of kth moment Not sure why the second integral is finite. Here is what I got in Mathematica with $k=20$. Integrate[Abs[x + 3]^k*(k + 1)^(k + 1)/x^(2 (k + 2)), {x, 1, [Infinity]}] // N 1.69501*10^38 Jun 19 comment \$X