End Behavior and Vertical Asymptotes for Rational Functions

Question

Find the equation for the vertical asymptote and state the end behavior of the graph around the vertical aymptote.

y = 1/(x² +3)

Answer 1

x^2>=0 for any real x

so then x^2 +3 >= 3 <--- please label this inequality ALPHA

and there are no vertical asymptotes

Moreover, f(0) = 1/3

Therefore, the function shall ALWAYS be positive since there are

no solutions to 1/(x^2+3) = 0

as x-->infinity, the function vanishes

as x--> negative infinity the function vanishes

per inequality ALPHA, 1/(x^2+3) <= 1/3

so the range is (0,1/3]

yes, (0,1/3) is the y-intercept and Absolute max