for example : $\lim_{n \to \infty} (1 + \frac1n)^n$

many thought this was equal to $1$ when they first encountered it.

the classical mistake here is that the theorem of composition of limits is applied to a function that isn't fixed but varies depending on $n$.

any others like this ?


closed as primarily opinion-based by rschwieb, Rob Arthan, Crostul, Leucippus, user223391 Mar 16 '18 at 1:02

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ mathoverflow.net/questions/23478/… $\endgroup$ – angryavian Mar 15 '18 at 21:43
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    $\begingroup$ You are conflating fame with ubiquity. $\endgroup$ – Rob Arthan Mar 15 '18 at 21:55
  • $\begingroup$ The limit of the expression you have written is $(1+\frac{1}{n})^n$ and I do not believe anybody will take it for 1. $\endgroup$ – user Mar 15 '18 at 21:56
  • $\begingroup$ @user that was a typo. thanks for noting. $\endgroup$ – rapidracim Mar 15 '18 at 21:57
  • $\begingroup$ What evidence do you have for your claim that many people think this limit is $1$ when they first encounter it? $\endgroup$ – Rob Arthan Mar 15 '18 at 22:01

Monty Hall Problem:

"Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

Many would thought it makes no difference whether you switch or not but the answer is switching is a better strategy if you compute the probability.

  • $\begingroup$ Assuming, of course, that you are not dealing with Evil Monty who, after seeing everybody telling us that we should switch, decides to only open an empty door whenever you initially pick the door with the price. $\endgroup$ – Bram28 Mar 15 '18 at 21:43

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