Question

Find Lim x tends to a (ctgx-ctga)/x-a

Answer 1

your question is -> find $\displaystyle\lim_{x\to a}\frac{cotx - cota}{x-a}$

[note : cotangent is written as cot or ctg ]

first of all, find form of limit ,

putting x = a , (cota - cota)/(a - a) = 0/0

hence, form of limit is 0/0.

we can apply L - Hospital rule

so, differentiating numerator and denominator individually with respect to x.

i.e., $\displaystyle\lim_{x\to a}\frac{-cosec^2x-0}{1-0}$

now putting x = a,

we get, -cosec²a

hence, $\displaystyle\lim_{x\to a}\frac{cotx - cota}{x-a}$ = -cosec²a