Asymptotes

[hdancarey]

show that x=1 and y=2x+3 are asymptoes for f(x)=(2x**2+x-1)/(x-1). I understand x=1 but do not understand how the y=2x+3 was determined.

 

[Deleted member 4993]

show that x=1 and y=2x+3 are asymptoes for f(x)=(2x**2+x-1)/(x-1). I understand x=1 but do not understand how the y=2x+3 was determined.

For a quick review, go to:

http://www.purplemath.com/modules/asymtote.htm

 

[galactus]

The 2x+3 is called an oblique asymptote. When we long divide $\displaystyle \frac{2x^{2}+x-1}{x-1}$, we get

$\displaystyle \frac{2}{x-1}+\underbrace{2x+3}_{\text{oblique asymptote}}$

As we can see, the oblique asymptote is a line the function approaches as $\displaystyle x\to \pm{\infty}$

See the graph I included.