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How do i solve the below without using L'hopital rule. The final answer obtained is $2/3$

$$\lim_{n\to\infty}\frac{\displaystyle{\cot\frac{2}{n}+n\csc\frac{3}{n^2}}}{\displaystyle{\csc\frac{3}{n}+n\cot\frac{2}{n^2}}}$$

How do i go about solving using limit of $$\lim_{x\to 0}\frac{\sin x}{x}=0,\\ \lim_{x\to 0}\frac{\tan x}{x}=1$$

I was shown to this step but how do i get to it? enter image description here

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i trying to solve it using limit of x goes to 0, Sinx/x =1 and limit of x goes to 0 tanx/x=1 –  grenn Feb 26 at 6:45
    
i know the limits to use are above but can't understand the trick to using it –  grenn Feb 26 at 6:59
    
Sorry ! I missed that. I change my answer. –  Claude Leibovici Feb 26 at 7:34
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1 Answer 1

Hint

All the arguments of the trigonometric functions are very small when $n$ is large. So, taking into account what you are supposed to use, rewrite you expression replacing $\csc(x)$ by $1/ \sin(x)$ and $\cot(x)$ by $1/ \tan(x)$ and, taking into account the limits you know, replace $\sin(x)$ by $x$ and $\tan(x)$ by $x$ with the proper corresponding arguments.

When this is done, reduce to same denominator both numerator and denominator of the expression and notice that $n^2$ is much larger than $n$. You sill arrive to your result.

I am sure you can take from here.

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haven't been taught this method yet so i'm using this "solving using limit of x goes to 0, Sinx/x =1 and limit of x goes to 0 tanx/x=1 " –  grenn Feb 26 at 7:17
    
how do i derive to that step as shown from my latest update. –  grenn Feb 26 at 9:37
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