COMEDK · Maths · 16. Limits

The value of \(\lim _{x \rightarrow 0}\left(\dfrac{a^{x}+b^{x}+c^{x}}{3}\right)^{2 / x},(a, b, c>0)\) is

A \((a b c)^{3}\)
B \(a b c\)
C \((a b c)^{1 / 3}\)
D None of these

Verified Solution

Answer & Solution

Correct Answer

(D) None of these

Step-by-step Solution

Detailed explanation

As \(x \to 0\), the base \(\dfrac{a^x+b^x+c^x}{3} \to 1\) and exponent \(\dfrac{2}{x} \to \infty\), giving \(1^\infty\) indeterminate form. Using \(\lim f(x)^{g(x)} = e^{\lim g(x)[f(x)-1]}\):…

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.

Same subject

Explore more questions on app

From COMEDK

Match the Coordination compounds given in Column I with their characteristic features listed in Column II.

S.No.	Coordination compounds	S.No.	Characteristic features
W	\([Co(NH_3)_5Cl]Cl_2\)	P	Oxidation state \(= +3\), Configuration \(= d^5\), \(\mu = 5.92\) BM
X	\(K_4[Mn(CN)_6]\)	Q	Oxidation state \(= +3\), Configuration \(= d^3\), \(\mu = 3.87\) BM
Y	\([CrCl_2(py)_3]\)	R	Oxidation state \(= +3\), Configuration \(= d^6\), \(\mu = 0\) BM
Z	\(Cs[FeCl_4]\)	S	Oxidation state \(= +2\), Configuration \(= d^5\), \(\mu = 1.732\) BM

COMEDK 2023 Medium

Explore more questions on app