<s>[INST] Correct way to solve a limit. I have the following limit,
$$\lim_{n\rightarrow \infty}\left \{ \frac{\left ( n+1 \right )\left ( n+2 \right )...3n}{n^{2n}} \right \}^{\frac{1}{n}}$$
My procedure of solving (which is wrong).
Step 1: I break up the expression in the following manner,
$$\lim_{n\rightarrow \infty }\left  \{ \left ( 1+\frac{1}{n} \right )\left ( 1+\frac{2}{n} \right )...\left ( \frac{3n}{n} \right ) \right  \}^{\frac{1}{n}}$$
Step 2: I apply the limits,
$$\left \{ \left ( 1+\frac{1}{\infty } \right ) \left ( 1+\frac{2}{\infty } \right )...\left ( 2+\frac{1}{\infty } \right )...3\right \}^{\frac{1}{\infty }}$$
which makes it, $$\left \{ 1^{\infty } *2^{\infty }*3\right \}^{0}$$
I am stuck here and could not proceed. Can somebody please help me out?
 [/INST] Try this: 
$$\lim_{n\rightarrow \infty} \left(\dfrac{n+1}{n}\cdot\dfrac{n+2}{n}\cdot\cdot\cdot\dfrac{3n}{n}\right)^{\dfrac{1}{n}}=\lim_{n\rightarrow \infty} \left(\left(1+\dfrac{1}{n}\right)\cdot\left(1+\dfrac{2}{n}\right)\cdot\cdot\cdot3\right)^{\dfrac{1}{n}}=\lim_{n\rightarrow \infty}\left(1+\dfrac{1}{n}\right)^{\dfrac{1}{n}}\cdot\lim_{n\rightarrow \infty}\left(1+\dfrac{2}{n}\right)^{\dfrac{1}{n}}\cdot\cdot\cdot\lim_{n\rightarrow \infty}3^{\dfrac{1}{n}}=1\cdot 1\cdot1\cdot\cdot\cdot1=1$$
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