Question

Cho hàm số \(y=f\left(x\right)=x+1+\dfrac{1}{x-1}\) có đồ thị (C) và đường thẳng y = x + 1 (Hình 15).

Tìm \(\lim\limits_{x\rightarrow+\infty}\left[f\left(x\right)-\left(x+1\right)\right];\lim\limits_{x\rightarrow-\infty}\left[f\left(x\right)-\left(x+1\right)\right]\).

Answer 1

Ta có: \(\left\{ \begin{array}{l}\mathop {\lim }\limits_{x \to  + \infty } \left[ {f\left( x \right) - \left( {x + 1} \right)} \right] = \mathop {\lim }\limits_{x \to  + \infty } \frac{1}{{x - 1}} = 0\\\mathop {\lim }\limits_{x \to  - \infty } \left[ {f\left( x \right) - \left( {x + 1} \right)} \right] = \mathop {\lim }\limits_{x \to  - \infty } \frac{1}{{x - 1}} = 0\end{array} \right.\)