We know that:
$\lim_{n\to \infty} \frac{\infty}{\infty} = \text{indeterminate}$
And that:
$\lim_{n\to \infty} \frac{n}{n} = 1$
How can I easily explain the difference to first-year university students?
$\endgroup$ 42 Answers
$\begingroup$You write
We know that: $\lim_{n\to \infty} \frac{\infty}{\infty} = indeterminate$
but I'd say that we don't know that at all. (Indeed, it makes no sense).
What we do know is that if $$ \lim_{n \to \infty} f(n) = \infty $$ and $$ \lim_{n \to \infty} g(n) = \infty $$ then $$ \lim_{n \to \infty} \frac{f(n)}{g(n)} $$ cannot be directly computed with a quotient rule.
I hate to say this, but I think your problem is not with "explaining this to first-year university students", but rather with knowing and understanding the main theorems yourself before trying to explain them to others.
$\endgroup$ 4 $\begingroup$I may first give an example : finding limit $$ \lim_{x \rightarrow \infty} \frac{1+x}{x} $$
When we use straightforward approach, we get $$ \frac{\infty+1}{\infty} = \frac{\infty}{\infty} $$ In the process of investigating a limit, we know that both the numerator and denominator are going to infinity.. but we dont know the behaviour of each dynamics. But if we investigate further we get : $$ 1 + \frac{1}{x} $$ Some other examples :
- Numerator might get larger than denomenator exactly $m$ times. The limit will be $m$ : for example $\lim \frac{mx}{x}$. Or the opposite : for example $\lim \frac{x}{mx}$
- The numerator gets way too large than denominator : $\lim \frac{x^2}{x}$ , the limit is clearly $\infty$. Or the opposite : $\lim \frac{x}{x^2}$.
Thanks.
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