Question

find the unknown constant if the given function is continuous

Answer 1

given f(X) = cos5x-1/√(5x²+1) -1

when X ≠0
for function being continuous
lim x----->0. cos5x-1/√(5x²+1) -1 = k

cos2t = 1-2 sin²t
cos2t-1 = -2sin²t
thus cos5x-1 = -2sin²5x/2

now rationalize

lim x-----> 0
-(2sin²(5x/2))*(√(5x²+1) +1)/(√(5x²+1) -1)*(√(5x²+1) +1)
= lim x----->0
-(2sin²(5x/2))*(√(5x²+1) +1)/5x²+1-1
= lim x----->0
-(2sin²(5x/2))*(√(5x²+1) +1)/5x²

lim t------> 0 sint /t = 1

thus lim x---->0
-2(√(5x²+1) +1) (sin²(5x/2)/(5x/2)²) *5/4

now put X = 0
= -2 *2*5/4 = -5 = k