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TIFR Mathematics 2023 | Part A | Question: 4

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Consider the function $f:(0, \infty) \rightarrow(0, \infty)$ given by $f(x)=x e^{x}$. Let $L:(0, \infty) \rightarrow(0, \infty)$ be its inverse function. Which of the following statements is correct?

  1. $\displaystyle{}\lim _{x \rightarrow \infty} \frac{L(x)}{\log x}=1$.
  2. $\displaystyle{}\lim _{x \rightarrow \infty} \frac{L(x)}{(\log x)^{2}}=1$.
  3. $\displaystyle{}\lim _{x \rightarrow \infty} \frac{L(x)}{\sqrt{\log x}}=1$.
  4. None of the remaining three options is correct.

     

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