Friday, 2 December 2016

limits - By letting m=frac1n find limnrightarrowinftyntanleft(frac1nright)



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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