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Question Number 92056 by john santu last updated on 04/May/20

$$\underset{x\to 0^{+}}{\lim }\frac{\ln \left(x\right)}{\cot x}$$

Commented by Prithwish Sen 1 last updated on 05/May/20

$$\underset{\mathrm{x}\to 0}{\lim }\mathrm{x}.\frac{\underset{\mathrm{k}=1}{\sum ^{\mathrm{\infty }}}{(-1)}^{\mathrm{k}+1}\frac{{(\mathrm{x}-1)}^{\mathrm{k}}}{\mathrm{k}}}{1-\underset{\mathrm{k}=1}{\sum ^{\mathrm{\infty }}}\frac{2^{2\mathrm{k}}}{\left(2\mathrm{k}\right)!}\mid {\mathrm{B}}_{2\mathrm{k}}\mid {\mathrm{x}}^{2\mathrm{k}}}=\frac{0}{1-0}=0$$

Commented by mathmax by abdo last updated on 04/May/20

$$f\left(x\right)=\frac{lnx}{cotanx}\Rightarrow f\left(x\right)=ln\left(x\right).tanx=\frac{sinxlnx}{cosx}$$$$sinxlnx\sim xln\left(x\right)\to 0(x\to 0)andcosx\sim 1\Rightarrow$$$${lim}_{x\to 0^{+}}f\left(x\right)=0$$

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