How do you find the vertical, horizontal or slant asymptotes for #f(x)=(2x-3)/(x^2+2)#?

Question

Madelyn Anderson · Accepted Answer

#"horizontal asymptote at "y=0#

The denominator of f(x) cannot be zero as this would make f(x) undefined. Equating the denominator to zero and solving gives the values that x cannot be and if the numerator is non-zero for these values then they are vertical asymptotes.

#"solve " x^2+2=0rArrx^2=-2#

#"this has no real solutions hence there are no "# #"vertical asymptotes"#

#"horizontal asymptotes occur as "#

#lim_(xto+-oo),f(x)toc" ( a constant)"#

divide terms on numerator/denominator by the highest power of x,that is #x^2#

#f(x)=((2x)/x^2-3/x^2)/(x^2/x^2+2/x^2)=(2/x-3/x^2)/(1+2/x^2)#

as #xto+-oo,f(x)to(0-0)/(1-0)#

#rArry=0" is the asymptote"#

Slant asymptotes occur when the degree of the numerator > degree of the denominator. This is not the case here hence there are no slant asymptotes. graph{(2x-3)/(x^2+2) [-10, 10, -5, 5]}

Madelyn Anderson · Answer

undefined To find the vertical, horizontal, or slant asymptotes for the function $ f(x) = \frac{2x - 3}{x^2 + 2} $:

1. **Vertical Asymptotes:** Set the denominator equal to zero and solve for $ x $. These values will give you the vertical asymptotes, if any.

$$ x^2 + 2 = 0 $$

$$ x^2 = -2 $$

$$ \text{No real solutions, so there are no vertical asymptotes.} $$

2. **Horizontal Asymptotes:** Compare the degrees of the numerator and the denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is at $ y = 0 $. If the degrees are equal, divide the leading coefficients. If the degree of the numerator is greater, there is no horizontal asymptote.

$$ \text{Degree of numerator} = 1 $$
$$ \text{Degree of denominator} = 2 $$

$$ \text{Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is } y = 0. $$

3. **Slant Asymptote (if any):** Perform polynomial long division or synthetic division to divide the numerator by the denominator. The quotient obtained represents the equation of the slant asymptote.

$$ \text{Polynomial long division or synthetic division of } (2x - 3) \text{ by } (x^2 + 2) $$

$$ \text{Resulting in } 0 \text{ with a remainder of } (2x - 3). $$

$$ \text{Therefore, the slant asymptote is } y = 0. $$