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If $\{x\}$ denotes the fractional part of $x$, then the limit $\lim_{x\to 1}\dfrac{e^{[x]}-\{x\}-1}{\{x\}^2}$, where $[x]$ is the greatest integer part of $x$, is

1. 0
2. $\dfrac{1}{2}$
3. -2
4. None of these

Correct answer: None of these

Solution

As $x\to 1^-$, we have $[x]=0$ and $\{x\}=x$, so the expression becomes $\frac{1-x-1}{x^2}=-\frac{1}{x}$. This tends to $-1$, which is not among the listed options, so the correct choice is None of these.

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