Question:
If
f(x)=
begincases
(xlog(cos x))/(log(1+x²)), & x≠ 0
0, & x=0
endcases
then f(x) is
If
f(x)=
begincases
(xlog(cos x))/(log(1+x²)), & x≠ 0
0, & x=0 endcases then f(x) is
0, & x=0 endcases then f(x) is
Show Hint
Continuity requires function value = limit, but differentiability is a stronger condition.
Updated On: Mar 19, 2026
- continuous as well as differentiable at x=0
- continuous but not differentiable at x=0
- differentiable but not continuous at x=0
- neither continuous nor differentiable at x=0
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The Correct Option is B
Solution and Explanation
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.
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