If fx=x2lncosxln1+x2, x≠00, x=0, then fx is

(B) continuous but not differentiable at zero

(D) not continuous and not differentiable at zero

AP EAMCET · Maths · Continuity and Differentiability

If $f (x) = \{\begin{array}{cl} \frac{x^{2} \ln \cos x}{\ln (1 + x^{2})} & , x \neq 0 \\ 0 & , x = 0 \end{array}$ , then $f (x)$ is

A discontinuous at zero
B continuous but not differentiable at zero
C differentiable at zero
D not continuous and not differentiable at zero

Verified Solution

Answer & Solution

Correct Answer

(C) differentiable at zero

Step-by-step Solution

Detailed explanation

f'0+=limh→0f0+h-f0h =limh→0fhh =limh→0lncoshln1+h2×h2h2 ....1 {Multiplying Numerator & denominator by h} =limh→0h2ln1+h2×limh→0lncoshh2 =limh→0h2ln1+h2×limh→0-tanh2h {Using L'Hospital's…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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From AP EAMCET

Match the following?

property	metal
(a) Element with highest second ionization enthalpy $(Δ_{i} H_{2})$	$(i) Co$
(b) Element with highest third ionization enthalpy $(Δ_{i} H_{3})$	$(ii) Cr$
(c) $M$ in $[M (CO)_{6}]$	$(iii) Cu$
(d) Element with highest heat of atomization $(Δ_{a} H)$	$(iv) Zn$
	$(v) Ni$

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