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I feel like there is something I am missing here. Is this as easy as it looks? Is the limit infinity? Or should I do L'hopital's rule?

With L'hopital I get 1/1 which is just 1.

Elsa
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3 Answers3

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That will work. You could also to do the following:

$$\lim_{n\to\infty}\frac n{n+1}=\lim_{n\to\infty}{n\cdot 1/n\over (n+1)\cdot 1/n=}=\lim_{n\to\infty}\frac1{1+1/n}=1.$$

Tim Raczkowski
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Hint:

$${ n \over n+1} = 1 - {1 \over n + 1}$$

Now, what's the limit of the right hand side as $n \to \infty$?

Simon S
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Convert it into a middle school word problem and think. The fraction $n/(n+1)$ means $n$ parts of a thing divided into $n+1$ equal parts. Day 1 I eat piece of a pizza that was sliced into two equal parts; on day 2 I eat 2 parts from a pizza sliced into 3 equal parts; on day 99 I would be eating 99 pieces from a pizza sliced into 100 equal parts. Eventually (limit) on a single day how many pizzas would I be eating? Not really a calculus problem needing L'Hopital's rule.

P Vanchinathan
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  • I feel like this is most like Simon S.'s answer, except with more words and more pizza. Mmm… – Akiva Weinberger Apr 16 '15 at 01:14
  • @columbus8myhw: A learner who thinks the limit could be infinity and thinks L'Hopital's rule needs to be applied should be told the problem is much simpler, and pizza is a way, I thought; if you don't like pizza's you can reformuate that into pancakes, or dosas ! – P Vanchinathan Apr 16 '15 at 01:25
  • Hey, no, I'm not saying this is bad in any way; sorry if it came out like that. – Akiva Weinberger Apr 16 '15 at 01:26
  • Mm, pizza. The Homer Simpson School of Mathematics. – Simon S Apr 16 '15 at 18:18