Need Math Help~! Pre-Calc !?

Question

ok , here is the question:

1) Does this function's graph intersect its horizontal asymptote?

g(x) = x / x ² + 1

2) Does this function's graph have more than one horizontal asymptote?

g(x) = | x - 1 | / x² - 4

3) Prove that the graph of every odd function with domain all real numbers must pas through the origin.

4) Show algebraically that -1 < 3x² - 1 / 2x² + 1 < 1.5


any help at all is appreciated, ~! thanks a lot~!! ^-^

Answer 1

1) Yes Asymptote is y=0. and the function crosses this line at the origin.
2) No just one which is the y-axis
3)If the function f(x) is odd, then -f(x)=f(-x). Thus it ca be rotated 180 degrees and still have the same orientation. Hence it is symmetric with respect to the origin.

4.As x approaches +/- infinity , (3x^2-1)/(2x^2+1) --> 1.5. This can be proven by using L'Hospital's rule. So (3x^2-1)/(2x^2+1)< 1.5. But if x = 0 we have -1<-1<1.5 which is not true. If x =sqrt(3)/3 , we have -1<0<1.5 which is OK. If x = 1/3, then we have -1<-2/3 <1.5.

So the inequality is good for all real values of x except 0.