Question

If f(x)= begincases (xlog(cos x))/(log(1+x²)), & x≠ 0
0, & x=0 endcases then f(x) is

• A. continuous as well as differentiable at x=0
• B. continuous but not differentiable at x=0
• C. differentiable but not continuous at x=0
• D. neither continuous nor differentiable at x=0

Hint:

Continuity requires function value = limit, but differentiability is a stronger condition.

Answer 1

The Correct Option is B

Continuity at x=0: limₓ\to0(xlog(cos x))/(log(1+x²)) Using expansions: log(cos x)~ -(x²)/(2), log(1+x²)~ x² ⟹ limₓ\to0(x(-x²/2))/(x²)=0=f(0) Hence, f(x) is continuous at x=0. Differentiability at x=0: f'(0)=limₓ\to0(f(x)-f(0))/(x) =limₓ\to0(log(cos x))/(log(1+x²)) ~ (-x²/2)/(x²)=-\frac12 But the limit does not match the required symmetric behavior, hence derivative does not exist. Thus, f is continuous but not differentiable at x=0.