Question

How do you find the vertical, horizontal or slant asymptotes for `(sqrt(x^4 + 6x^2 + 9)) / (2x^2 - 10)`?

Answer 1

Vertical asymptote is at `x= +-sqrt5`
Horizontal asymptote: `y=0.5`, slant asymptote:absent

Explanation:

`f(x)=sqrt(x^4+6x^2+9)/(2x^2-10)` or

`f(x)=sqrt((x^2+3)^2)/(2( x^2-5)` or

`f(x)=((x^2+3))/(2( x^2-5)`

Vertical asymptote is at `x^2-5=0 or x^2=5 or x=+-sqrt5`

Horizontal asymptote: The degree of numerator and denominator

is same i.e `2` , so we have a horizontal asymptote at

`y =` (numerator's leading coefficient) / (denominator's leading

coefficient) `:. y= 1/2=0.5`

Slant asymptote: Since numerator's degree is not greater , we

have no slant asymptote.

Vertical asymptote is at `x= +-sqrt5`

Horizontal asymptote: `y=0.5`, slant asymptote:absent

graph{(sqrt(x^4+6x^2+9))/(2x^2-10) [-10, 10, -5, 5]} [Ans]