Question

If f(x) = `{{:((log_(sin|x|) cos^2x)/(log_(sin|3x|) cos  x/2), |x| < π/3; x ≠ 0),(k, x = 0):}`, then value of k for which f(x) is continuous at x = 0 is ______.

Options

• 7.00

• 8.00

• 9.00

• 10.00

Answer 1

If f(x) = `{{:((log_(sin|x|) cos^2x)/(log_(sin|3x|) cos  x/2), |x| < π/3; x ≠ 0),(k, x = 0):}`, then value of k for which f(x) is continuous at x = 0 is 8.00.

Explanation:

k = `lim_(x→0) (2logcosx)/(logsinx).(logsin3x)/(logcos  x/2)`

= `2lim_(x→0)(log(1 + cosx - 1))/(log(1 + cos  x/2 - 1)) (logsin3x)/(logsinx)`

= `2lim_(x→0)(-2sin^2  x/2)/(-2sin^2  x/4).(log(sinx(3 - 4sin^2x)))/(logsinx)`

= `2lim_(x→0) (x^2/4)/(x^3/16)(1 + (log(3 - 4sin^2x))/(logsinx))`

= 2.41

= 8