Find the limit of f(x)=(x-sin(x))/(x-tan(x)) as x tends to 0.
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The value of `lim_(x->0) (x-sin x)/(x-tan x)` has to be determined.
If x is substituted with 0, the value of the expression is 0/0 which is indeterminate. This allows the use of l'Hopital's rule with the substitution of the numerator and denominator with their derivatives.
=> `lim_(x->0) (1-cos x)/(1-sec^2 x)`
=> `lim_(x->0) (1-cos x)/((1-sec x)(1+sec x))`
=> `lim_(x->0) ((1-cos x)(cos^2x))/((cos x-1)(cos x+1))`
=> `lim_(x->0) -(cos^2x)/(cos x+1)`
substitute x = 0
The value of `lim_(x->0) (x-sin x)/(x-tan x) = -1/2`
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