How do you find the vertical, horizontal or slant asymptotes for #(2x+sqrt(4-x^2)) / (3x^2-x-2)#?

Question

Lily Anderson · Accepted Answer

#-2<=x<=2#. y-intercept: x-intercept: #-2/sqrt5#. Dead ends #(-2, -1/6) and (2, 1)#. .Vertical asymptote: x = 1. See the explanation and graphs, for clarity.

To make y real, #-2<=x<=2#. Accordingly, we reach dead ens at #(-2, -1/6) and (2, 1)#. y-intercept ( x = 0 ) is 1 and x-intercept ( y = 0 ) is #-2/sqrt5#. Note that the curve does not reach the x-axis for positive intercept. Also, #y = (1/x)(2+sqrt(1-2/x^2))/(3-1/x+2/x^2) to 0# as # x to +-oo# does not happen, as #|x|<=2#, Now, #y =(2x+sqrt(4-x^2))/((x-1)(3x+2)) to +-oo# as x to 1 and x to -2/3# The first is OK. but, the second is negated by #x >=-2#. Important note: The convention #sqrt(4-x^2)>=0 ( unlike (4-x^2)^(1/2)=+-sqrt(4-x^2))# troubled me in making my data compatible with that in the graph. I had to take negative root, breaking the convention. graph{y(x-1)(3x-2)-2x-sqrt(4-x^2)=0 [-10.17, 10.17, -5.105, 5.065]} graph{y(x-1)(3x-2)-2x-sqrt(4-x^2)=0 [-20.34, 20.34, -10.19, 10.15]}

Xavier Alonso · Answer

undefined To find the vertical, horizontal, or slant asymptotes of the function $ \frac{2x + \sqrt{4 - x^2}}{3x^2 - x - 2} $, we'll analyze the behavior of the function as $ x $ approaches different values.

1. Vertical Asymptotes:
   Vertical asymptotes occur where the denominator of the function becomes zero, but the numerator does not. So, we set the denominator equal to zero and solve for $ x $:
   $$ 3x^2 - x - 2 = 0 $$
   Solving this quadratic equation, we find the roots:
   $$ x = \frac{1 \pm \sqrt{1 + 24}}{6} $$
   $$ x = \frac{1 \pm \sqrt{25}}{6} $$
   $$ x = \frac{1 \pm 5}{6} $$
   $$ x = -\frac{2}{3}, \, \frac{1}{3} $$
   So, the vertical asymptotes are at $ x = -\frac{2}{3} $ and $ x = \frac{1}{3} $.

2. Horizontal Asymptote:
   To find the horizontal asymptote, we examine the behavior of the function as $ x $ approaches positive or negative infinity. We'll perform long division or divide the leading terms of the numerator and denominator:
   $$ \lim_{x \to \pm \infty} \frac{2x + \sqrt{4 - x^2}}{3x^2 - x - 2} $$
   Since the degree of the numerator is 1 and the degree of the denominator is 2, the horizontal asymptote is at $ y = 0 $.

3. Slant Asymptote:
   To find the slant asymptote, we perform polynomial long division or use synthetic division if the degree of the numerator is one greater than the degree of the denominator. If the degree of the numerator is two or more greater, then no slant asymptote exists.

In this case, since the degree of the numerator is less than or equal to the degree of the denominator, there is no slant asymptote.