Let `f(x)={((log)_ecosx)/(1+x^2 4-1),x >0(e^(sin4x)-1)/((log)_e(1+tan2x),x<0` Find the value of `f(0)` which makes the function continuous at `x=0`

lim_(x rarr0)(sqrt(1+sin3x)-1)/(log(1+tan2x))

let f(x)=(ln cos x)/((1+x^(2))^((1)/(4))-1) if x>0 and f(x)=(e^(sin4x)-1)/(ln(1+tan2x)) if x<0 Is it possible to difine f(0) to make the function continuous at x=0. If yes what is.the value of f(0), if not then indicate the nature of discontinuity.

e^(tan^(-1)x)log(tan x)

Integration of (1)/(1+((log)_(e)x)^(2)) with respect to (log)_(e)x is ((tan^(-1)((log)_(e)x))/(x)+C(b)tan^(-1)((log)_(e)x)+C(c)(tan^(-1)x)/(x)+C(d) none of these

Find k, if possible, so that f (x)= [{:((ln (2- cos 2x))/(ln ^(2) (1+ sin 3x)),,, x lt 0),(k,,, x=0),((e ^(sin 2x )-1)/(ln (1+ tan 9x)),,, x gt 0):} is continious at x=0.

'f(x)={(e^(^^)x-1)/(log(1+2x)),|x!=0Kx=0at

The differential coefficient of f((log)_e x) with respect to x , where f(x)=(log)_e x , is (a)x/((log)_e x) (b) 1/x(log)_e x (c)1/(x(log)_e x) (d) none of these

The primitive of the function f(x)=(1-(1)/(x^(2)))a^(x+(1)/(x)),a>0 is (a^(x+(1)/(x)))/((log)_(e)a) (b) (log_(e)adot a^(x+(1)/(x))(c)(a^(x+(1)/(x)))/(x)(log)_(e)a(d)x(a^(x+(1)/(x)))/((log)_(e)a)

int((1+(log)_(e)x)^(2))/(1+(log)_(e)x^(x+1)+((log)_(e)x^(sqrt(x)))^(2))dx=