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TIFR2021-Maths-A: 7

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Let $A$ be the set of all real numbers $\lambda \in [0,1]$ such that

$$\displaystyle\lim_{p\rightarrow 0}\frac{\log(\lambda2^p+(1-\lambda)3^p)}{p}=\lambda \log2+(1-\lambda)\log3$$

Then

  1. $A=\{0,1\}$
  2. $A=\{0,\frac{1}{2},1\}$
  3. $A=\{0,\frac{1}{3},\frac{1}{2},\frac{2}{3},1\}$
  4. $A=[0,1]$
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