How do you find vertical, horizontal and oblique asymptotes for #(8x^2-5)/(2x^2+3)#?

Question

James Allen · Accepted Answer

#"horizontal asymptote at "y=4#

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 "2x^2+3=0rArrx^2=-3/2#

#"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)=((8x^2)/x^2-5/x^2)/((2x^2)/x^2+3/x^2)=(8-5/x^2)/(2+3/x^2)#

#"as "xto+-oo,f(x)to(8-0)/(2+0)#

#rArry=4" is the asymptote"#

#"Oblique asymptotes occur when the degree of the "# #"numerator is greater than the degree of the denominator"# #"this is not the case here hence there are no oblique"# #"asymptotes"# graph{(8x^2-5)/(2x^2+3 [-10, 10, -5, 5]}

Elijah Allen · Answer

undefined To find the vertical asymptotes of a rational function, set the denominator equal to zero and solve for $x$.

For the function $\frac{8x^2 - 5}{2x^2 + 3}$:

$2x^2 + 3 = 0$

Solving for $x$, we get:

$x^2 = -\frac{3}{2}$

Since there are no real solutions for $x^2 = -\frac{3}{2}$, there are no vertical asymptotes.

To find the horizontal asymptote, compare the degrees of the numerator and denominator:

The degree of the numerator is 2 and the degree of the denominator is also 2.

So, divide the leading coefficient of the numerator by the leading coefficient of the denominator:

$\frac{8}{2} = 4$

Therefore, the horizontal asymptote is $y = 4$.

To find the oblique asymptote, perform polynomial long division or synthetic division to divide the numerator by the denominator. In this case, since the degrees of the numerator and denominator are the same, there is no oblique asymptote.

Greyson Brown · Answer

undefined To find the vertical asymptotes, set the denominator equal to zero and solve for x. For horizontal asymptotes, compare the degrees of the numerator and denominator polynomials. 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 coefficients of the polynomials to find the horizontal asymptote. For oblique asymptotes, perform long division or synthetic division to divide the numerator by the denominator. The quotient represents the equation of the oblique asymptote.