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Question Number 203471 by DEGWE last updated on 19/Jan/24

Answered by MathematicalUser2357 last updated on 20/Jan/24

$$=\frac{\underset{x\to 0^{+}}{\lim }{x}^{\mathrm{cosec}x+\cot x}}{\underset{x\to 0^{+}}{\lim }(\mathrm{cosec}x+\cot x)}$$$$=\frac{\underset{x\to 0^{+}}{\lim }{x}^{\frac{1}{\sin x}+\frac{1}{\tan x}}}{\underset{x\to 0^{+}}{\lim }(\frac{1}{\sin x}+\frac{1}{\tan x})}$$$$=\frac{\underset{x\to 0^{+}}{\lim }{x}^{\frac{1+\frac{\sin x}{\tan x}}{\sin x}}}{\underset{x\to 0^{+}}{\lim }\frac{1+\frac{\sin x}{\tan x}}{\sin x}}$$$$=\frac{\underset{x\to 0^{+}}{\lim }{x}^{\frac{1+\cos x}{\sin x}}}{\underset{x\to 0^{+}}{\lim }\frac{1+\cos x}{\sin x}}$$$$=\frac{0}{1}$$$$=0$$

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