Why don't we use undefined forms to find limits of infinity?

Question

I came across a couple of simple limits of infinity and solved them using the normal subsequence method. Yet, I also tried solving them by substituting the undefined form $\frac{1}{0}$, and although this is not logical or rigorous, I managed to obtain the correct answers for the limit as $n$ goes to infinity for the reciprocal of $n$ and the limit as $n$ goes to infinity of the recpirocal of $3n-1$. However, I'm not certain how to prove that this method works or doesn't work for all cases of limits of infinity, and I was wondering if there is a way to do so. Also, I was wondering if this method can be applied even if it isn't analytically valid.

So, my questions is:

1) Is it logical to substitute $n=1/0$ for cases where we have to find the limits of such functions such as the limit as $n$ goes to infinity of $\frac{1}{n}$ , even if we obtain the correct answer when we substitute it?

Answer 1

You have that for any $a \in \mathbb{R}$

$$\lim_{n \to + \infty} \left( 1+ \frac{a}{n} \right)^n = e^a$$

But if you substituting $n$ by $\frac{1}{0}$ was valid, you would get

$$1^{\frac{1}{0}} = e^a$$

So even if you could makes sense of $1^{\frac{1}{0}}$, it would still lead to a contradiction

Answer 2

$n=\frac 10=?$

$\frac 10$ is not defined and so it is not logical to substitute $n=\frac 10$.

This just a coincidence that when you substitute $n=1/0$ in $\lim_{n \to 0} \frac 1n $ you get 0.

I want to give an example(this is not what you are exactly asking)

$x\in\mathbb{R}$ then what is $\frac{x+1}{x+1}$ the answer is $1$ but this is not completely valid because if $x\ne1$ then only $\frac{x+1}{x+1}=1$

Answer 3

In a word, "no", it's not logical. Merely writing down true statements does not constitute doing mathematics. Lots of incorrect procedures lead to correct conclusions: Just perform an arbitrary set of actions and incant arbitrary assertions, then write down a truism. (Look at the entrails of this piece of roadkill. Further, Jupiter is Cogitating in the House of Uranus. Therefore $\lim\limits_{n \to \infty} \frac{1}{n} = 0$. You get the idea....)

In the particular case of evaluating $\lim\limits_{n \to +\infty} \frac{1}{n} = 0$, your intuition presumably runs something like this: $\frac{1}{0} = +\infty$, so $n \to +\infty = \frac{1}{0}$, and taking reciprocals of each side gives $\lim\limits_{n \to +\infty} \frac{1}{n} = \frac{1}{+\infty} = 1/(1/0) = 0$.

The mathematical alternatives are:

• One is invoking theorems stronger than the conclusion one wishes to deduce.

• One is talking nonsense. For example, $\frac{1}{0}$ has no meaning as a quotient of real numbers, even in the extended real number system $[-\infty, +\infty] := \mathbf{R} \cup \{-\infty, +\infty\}$. (It's a common misconception that $\lim\limits_{x \to 0} \frac{1}{x} = +\infty$. But even the true statement $\lim\limits_{x \to 0} \frac{1}{x^{2}} = +\infty$ is not an equation of real numbers; it's a way of expressing that a particular limit fails to exist, albeit in a special manner from which additional information can be extracted.)