Is $\lim_{n\to\infty}\frac{1}{1/n}$ infinity or undefined? As above, if you have 
$$
\lim_{n\rightarrow∞}\frac{1}{\frac{1}{n}}
$$
Do you simplify first to get $\lim_{n\rightarrow∞}n$, or do you apply the limit and get the undefined $\frac{1}{0}$?
 A: Take a look at this description for limit rules:
http://tutorial.math.lamar.edu/Classes/CalcI/LimitsProperties.aspx
Basically,
You cannot apply the limit to both the numerator and denominator because the limit of the denominator goes to zero, i.e: 
$$
\lim_{x \to a} \left[ {\frac{{f\left( x \right)}}{{g\left( x \right)}}} \right] = \frac{{\mathop \lim_{x \to a} f\left( x \right)}}{{\mathop\lim_{x \to a} g\left( x \right)}}{\rm{,}}\,\,\,\,\,{\rm{provided  }}\,\mathop {\lim }\lim_{x \to a} g\left( x \right) \ne 0
$$
So basically the answer should just be:
$$
\lim_{n \to \infty} n
$$
A: Strictly speaking, as $n$ goes to positive infinity, $\frac{1}{n}$ never really becomes zero, it's just in the constant process of getting arbitrarily close to it. What you're really doing is that you're dividing $1$ by a number that's getting closer to zero. The result of that is going to be a number that's getting larger and larger. A more proper way would be to algebraically manipulate the expression into a form that lets you more easily see what's going on with the limit as the variable approaches a particular value and then going by that make a decision as to whether the limit itself approaches a particular value. In your case, flip the fraction and simply observe that the limit mist go to positive infinity as $n$ itself goes to positive infinity:
$$
\lim_{n\to\infty}\frac{1}{\frac1n}=\lim_{n\to\infty}n=\infty.
$$
A: When we say that a limit is undetermined, it is just because we can't say right away if it exists or what is its value. In a sense, the undetermination is just a measure of our ignorance. For instance, we say that  0/0 is an undetermination because the result may vary depending on the numerator and the denominator. $\lim \frac{1/n}{1/n}$ and $\lim \frac{1/n^2}{1/n}$ both lead to 0/0 but the values of the limits are 1 and 0, respectively.
However, in your case, 1/0 is not an indetermination, the result is always $\infty$.
