How do you find the vertical, horizontal and slant asymptotes of: # f(x)= (3x + 5) /( x - 2)#?

Question

Elijah Allen · Accepted Answer

vertical asymptote x = 2
horizontal asymptote y = 3

The denominator of f(x) cannot be zero as this would be undefined. Equating the denominator to zero and solving gives the value that x cannot be and if the numerator is non-zero for this value of x then it is a vertical asymptote. solve : x - 2 = 0 → x = 2 is the asymptote Horizontal asymptotes occur as #lim_(xto+-oo),f(x)toc" (a constant)"# divide terms on numerator/denominator by x #((3x)/x+5/x)/(x/x-2/x)=(3+5/x)/(1-2/x)# as #xto+-oo,f(x)to(3+0)/(1-0)# #rArry=3" is the asymptote"# Slant asymptotes occur when the degree of the numerator > degree of the denominator. This is not the case here (both of degree 1 ) Hence there are no slant asymptotes. graph{(3x+5)/(x-2) [-20, 20, -10, 10]}

Madelyn Anderson · Answer

undefined To find the vertical asymptotes, horizontal asymptotes, and slant asymptotes of the function $ f(x) = \frac{3x + 5}{x - 2} $, follow these steps:

1. **Vertical Asymptotes**:
   Vertical asymptotes occur where the denominator of the rational function equals zero, but the numerator does not. Set the denominator equal to zero and solve for $ x $. The values of $ x $ obtained will be the equations of the vertical asymptotes.

2. **Horizontal Asymptotes**:
   Horizontal asymptotes occur when the degree of the numerator and denominator are the same. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is $ y = 0 $. If the degrees are equal, divide the leading coefficient of the numerator by the leading coefficient of the denominator to find the horizontal asymptote.

3. **Slant Asymptotes**:
   Slant asymptotes occur when the degree of the numerator is exactly one more than the degree of the denominator. To find the slant asymptote, perform polynomial long division of the numerator by the denominator. The quotient obtained will be the equation of the slant asymptote.

Apply these steps to find the vertical, horizontal, and slant asymptotes of the given function.