Question

Solve for this function
Lim x to 0 x/tanx

Answer 1

`\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`

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