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

question 168

Multiple Choice

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) \(\lim _ { x \rightarrow \theta ^ { - } } f ( x ) = - \infty , \lim _ { x \rightarrow \theta ^ { + } } f ( x ) = \infty , \lim _ { x \rightarrow \mathbb { Z } ^ { - } } f ( x ) = - \infty , \lim _ { x \rightarrow 2 ^ { + } } f ( x ) = \infty\)

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Definitions:

Comparative Advantage

The ability of a country, individual, company, or region to produce a good or service at a lower opportunity cost than a competitor, leading to more efficient international trade and allocation of resources.

Production Possibilities Line

A graphical representation that shows the maximum possible output combinations of two products or services an economy can achieve when all resources are fully and efficiently utilized.

Trading Possibilities Line

A graphical representation illustrating the maximum amount of one good that a country can trade for another, given fixed resources.

Comparative Advantage

The ability of a country or individual to produce a particular good or service at a lower opportunity cost than another entity, forming the basis for trade.