High School

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Solve for this function

Lim x to 0 x/tanx

Answer :

[tex]\displaystyle \lim_{x\to 0} ~~ \cfrac{\sin(x)}{x}=1\hspace{9em}\lim_{x\to 0} ~~\cfrac{1-\cos(x)}{x}=0 \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ \displaystyle \lim_{x\to 0} ~~ \cfrac{x}{\tan(x)} \\\\[-0.35em] ~\dotfill[/tex]

[tex]\cfrac{x}{\tan(x)}\implies \cfrac{x}{~~ \frac{\sin(x) }{\cos(x) } ~~}\implies \cfrac{x\cos(x)}{~~\sin(x) ~~}\implies \cfrac{1}{\frac{~~\sin(x) ~~}{x\cos(x)}}\implies \cfrac{1}{ ~~ \frac{\sin(x)}{x}\cdot \frac{1}{\cos(x)} ~~ } \\\\[-0.35em] ~\dotfill\\\\ \displaystyle \lim_{x\to 0} ~~ \cfrac{1}{ ~~ \frac{\sin(x)}{x}\cdot \frac{1}{\cos(x)} ~~ }\implies \cfrac{1}{1\cdot \frac{1}{\cos(0)}}\implies \cfrac{1}{1\cdot \frac{1}{1}}\implies \text{\LARGE 1}[/tex]

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