How do you find the vertical, horizontal or slant asymptotes for #f(x) = (4x^2 - 1 ) / (2x^2 + 5x - 12)#?

Question

Sophie Adams · Accepted Answer

vertical asymptotes # x = -4 , x = 3/2#
horizontal asymptote y = 2

When the denominator of a rational function tends to zero, vertical asymptotes occur. Let the denominator equal zero to find the equation or equations. solve : # 2x^2 + 5x - 12 = 0 → (2x-3)(x+4) = 0# #rArr x = - 4 " and " x = 3/2 " are the asymptotes " # Horizontal asymptotes occur as #lim_(x→±∞) f(x) → 0 # divide all terms on numerator/denominator by # x^2 # #((4x^2)/x^2 - 1/x^2)/((2x^2)/x^2 + (5x)/x^2 - 12/x^2) = (4-1/x^2)/(2+5/x-12/x^2)# now as x →∞ # 1/x^2 , 5/x " and " 12/x^2 → 0# #rArr y = 4/2 → y = 2 " is the asymptote " # There are no slant asymptotes in this case because the degree of the numerator is not greater than the degree of the denominator, which is the condition under which slant asymptotes occur. The function's graph is shown here: graph{(4x^2-1)/(2x^2+5x-12) [-10, 10, -5, 5]}

Christian Antunez · Answer

undefined To find the vertical asymptotes of $f(x) = \frac{4x^2 - 1}{2x^2 + 5x - 12}$, set the denominator equal to zero and solve for $x$.

$$2x^2 + 5x - 12 = 0$$

Factoring the quadratic equation or using the quadratic formula, we find the roots.

$$2x^2 + 8x - 3x - 12 = 0$$
$$2x(x + 4) - 3(x + 4) = 0$$
$$(2x - 3)(x + 4) = 0$$

This gives $x = \frac{3}{2}$ and $x = -4$. These are the vertical asymptotes of the function.

To find horizontal or slant asymptotes, we need to check the degrees of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is at $y = 0$. If the degree of the numerator is equal to the degree of the denominator, or greater by 1, then there is no horizontal asymptote, but there might be a slant asymptote.

In this case, both the numerator and denominator have the same degree (2), so we don't have a horizontal asymptote. To check for a slant asymptote, perform polynomial long division or synthetic division to divide the numerator by the denominator. If the quotient approaches a constant value as $x$ goes to positive or negative infinity, that constant value represents the equation of the slant asymptote.

After performing the division, if any, you'll find the equation of the slant asymptote.