find k if f(x) is continous at x=0. f(x)={(1-cos4x)/x^2 if x k if x=0 (√x)/√16+root of x-4 } Share with your friends Share 15 Vijay Kumar Gupta answered this Consider the following function. fx=1-cos4xx2, when x<0 =k, x=0 =x16+x-4 , when x>0 LHL = limx→0-fx=limx→0-1-cos4xx2 put x = 0-h, as x→0-, then h→0LHL = limh→01 - cos 4hh2 =limh→02 sin22hh2 = 2 limh→0sin 2h2h×2limh→0sin 2h2h×2 = 8 RHL =limx→0+fx= limx→0+x16+x-4 =limh→0 h16+h - 4 put x = 0+h, as x→0+, then h→0=limh→0 h16+h - 4×16+h + 416+h + 4=limh→0 h16+h + 416+h - 16=limh→0 16+h + 4 = 8 Also, f0 = kSince f is continuous at x = 0, thenLHL = RHL = f0⇒8 = 8 = k⇒k = 8 50 View Full Answer Pragati Mishra answered this for f(x) to be continuous at x=0 lim _ f (x)=f (0) x->0 lim _ f (x)=lim (1-cos 4x)/x^(2) x->0 x->0 =lim 2 sin^(2) 2x/x^(2) x->0 =lim (2*4) sin^(2) 2x/ (2^(2) * x^(2)) x->0 =8lim (sin 2x/2x)^(2) x->0 =8*1 [using lim sin x/x =1] x->0 =8 =f (0) =k Hence, k=8 2