Evalutation of the limit $\;\lim\limits_{x \to+ \infty}\left[x - x^2 \cdot \ln\left(1+ \frac 1 x\right)\right]$

Question

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

Answer 1

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

• Are all limits solvable without L'Hôpital Rule or Series Expansion