Evaluate the following limit: `lim_(x->0)(2x-sin x)/(tanx+x)`

Explanation

`(lim)_(x->0)(2x-sin x)/(tanx+x)`

`=(lim)_(x->0)((2x-sin x)/x)/((tanx+x)/x)`

`=(lim)_(x->0)((2x)/x-(sin x)/x)/((tanx/x)+x/x)`

`=(lim)_(x->0)(2-(sin x)/x)/((tanx/x)+1)`

`implies ((lim)_(x->0) 2-lim_(x to 0)(sin x))/(lim_(x to 0)(tanx)/x+lim_(x to 0)1)`

`implies lim_(x to 0) (2x-sin x)/(tan x+x)`

`implies (2-lim_(x to 0)(sin x)/x)/(lim_(x to 0)(tan x)/x+1)`

`=1/2`