Question

please solve it.....​

Answer 1

Answer:

Note first that tan(x - y) = (tan(x) - tan(y)) / (1 + tan(x)*tan(y)).

So tan(x) - tan(y) = tan(x - y) * (1 + tan(x)*tan(y)), and thus

lim(x->y)[(tan(x) - tan(y)) / (x - y)] =

lim(x->y)[(1 + tan(x)*tan(y)) * (tan(x - y) / (x - y))] =

lim(x->y)(1 + tan(x)*tan(y)) * lim(x->y)(tan(x - y) / (x - y)) =

(1 + tan^2(x)) * 1 = sec^2(x).

Edit: To evaluate lim(x->y)(tan(x - y) / (x - y)), let u = x - y.

Then u -> 0 as x -> y and so the limit becomes

lim(u->0)(tan(u) / u) = lim(u->0)((1/cos(u)) * (sin(u) / u)) =

lim(u->0)(1/cos(u)) * lim(u->0)(sin(u) / u) = 1*1 = 1,

where lim(u->0)(sin(u) / u) = 1 is a well-known limit.