By letting m=1n find
lim
I've played around with the algebra, but can't see how m fits in, apart from abbreviation. Is my working correct?
\dfrac{\tan{\frac{1}{n}}}{\frac{1}{n}}=\frac{0}{0}
L'Hopital's rule:
\dfrac{\sec^2m}{(-1/n^2 )}=\dfrac{-n^2}{\cos^2\frac{1}{n}}
Divide by -n^2 to get \dfrac{1}{\cos^2\frac{1}{n}} which is 1 as n tends to infinity because \cos0=1. I don't really get limits yet, but is this right?
Answer
Here are the steps
\lim\limits_{n\to\infty} n\tan\left(\frac{1}{n}\right)
Let m=\frac{1}{n}. Since n\to\infty, then m=\frac{1}{\infty}=0. So now we have
\lim\limits_{m\to 0} \frac{\tan\left(m\right)}{m}= \lim\limits_{m\to 0} \frac{\frac{d}{dm}\tan\left(m\right)}{\frac{d}{dm}m} = \lim\limits_{m\to 0} \sec^2\left(m\right) = \lim\limits_{m\to 0} \frac{1}{\cos^2\left(m\right)}=1
Looks to me like you do get limits. Keep it up.
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