Use limits to describe the behavior of the rational function near the indicated asymptote.
-$$f ( x ) = \frac { x + 1 } { x ^ { 2 } - 2 x }$$
Describe the behavior of the function near its vertical asymptotes.
A) $\lim _ { x \rightarrow \theta ^ { - } } f ( x ) = \infty , \lim _ { x \rightarrow \theta ^ { + } } f ( x ) = - \infty , \lim _ { x \rightarrow 2 ^ { - } } f ( x ) = \infty , \lim _ { x \rightarrow 2 ^ { + } } f ( x ) = - \infty$
B) $\operatorname { lim~ } _ { x \rightarrow \theta ^ { - } } f ( x ) = 0 , \lim _ { x \rightarrow \theta ^ { + } } f ( x ) = - \infty , \lim _ { x \rightarrow 2 ^ { - } } f ( x ) = - \infty , \quad \lim _ { x \rightarrow 2 ^ { + } } f ( x ) = \infty$
C) $\lim _ { x \rightarrow \theta ^ { - } } f ( x ) = \infty , \lim _ { x \rightarrow \theta ^ { + } } f ( x ) = - \infty , \lim _ { x \rightarrow 2 ^ { - } } f ( x ) = - \infty , \lim _ { x \rightarrow 2 ^ { + } } f ( x ) = \infty$
D) $\lim _ { x \rightarrow \theta ^ { - } } f ( x ) = - \infty , \lim _ { x \rightarrow \theta ^ { + } } f ( x ) = \infty , \lim _ { x \rightarrow \mathbb { Z } ^ { - } } f ( x ) = - \infty , \lim _ { x \rightarrow 2 ^ { + } } f ( x ) = \infty$

Question

Quizplus · Accepted Answer

First, factor the denominator: $x^2 - 2x = x(x - 2)$, revealing vertical asymptotes at $x = 0$ and $x = 2$. As $x$ approaches 0 from the left, the function goes to $\infty$, and from the right, it goes to $-\infty$. Approaching 2 from the left, the function goes to $-\infty$, and from the right, it goes to $\infty$.

Use Limits to Describe the Behavior of the Rational Function f(x)=x+1x2−2xf ( x ) = \frac { x + 1 } { x ^ { 2 } - 2 x }f(x)=x2−2xx+1​

Use Limits to Describe the Behavior of the Rational Function $f ( x ) = \frac { x + 1 } { x ^ { 2 } - 2 x }$