How do you find the asymptotes for #f(x) = (2x-1) / (x - 2)#?

Question

James Allen · Accepted Answer

Let rewrite this as follows #f(x)=(2x-4+3)/(x-2)=2+3/(x-2)# hence for #x->oo# , #f(x)->2# and for #x->2# , #f(x)->oo# Hence #y=2# is the horizontal asymptote and #x=2# the vertical asymptote

Anna Allen · Answer

undefined To find the asymptotes of the function $ f(x) = \frac{2x - 1}{x - 2} $, we need to identify vertical and horizontal asymptotes.

1. Vertical Asymptote:
The vertical asymptote occurs where the denominator of the function becomes zero. So, to find the vertical asymptote, set the denominator equal to zero and solve for $ x $.
$$ x - 2 = 0 $$
$$ x = 2 $$
Therefore, there is a vertical asymptote at $ x = 2 $.

2. Horizontal Asymptote:
To find the horizontal asymptote, we examine the behavior of the function as $ x $ approaches positive or negative infinity. To do this, we look at the degrees of the numerator and denominator.
$$ \text{Degree of numerator} = 1 $$
$$ \text{Degree of denominator} = 1 $$
Since the degrees of the numerator and denominator are equal, we look at the ratio of the leading coefficients:
$$ \frac{2}{1} = 2 $$
Therefore, the horizontal asymptote is $ y = 2 $.

In summary, for the function $ f(x) = \frac{2x - 1}{x - 2} $:
- There is a vertical asymptote at $ x = 2 $.
- There is a horizontal asymptote at $ y = 2 $.