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I was trying to evaluate the limit $$\lim_{x \to+ \infty} x - x^2 \cdot \ln\left(1+ \frac 1 x\right)$$ without using neither Taylor series nor De L'Hopital rule, but just with notable limits such as $\lim_{x \rightarrow 0} \frac {e^x - 1} x = 0$ or substitution.

I tried for a lot of times with different substitutions and notable limits, but I couldn't find any solution.

Can you give me some hints.

Thanks in advance

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HINT

We have that

$$x - x^2 \cdot \ln\left(1+ \frac 1 x\right)=\frac{\frac1x -\ln\left(1+ \frac 1 x\right)}{\frac1{x^2}}$$

then refer to

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