find all vertical and horizontal asymptotes f(x)=(3x+2)/(4x^2+2)

Question

anonymous · Answer

$$f(x)=\frac{ 3x+2 }{ 4x ^{2} +2}$$

anonymous · Answer

Is there a solution to $\large 4x^2+2=0 $  ???

anonymous · Answer

4x^2=-2   x^2=-1/2  x= radical 1/2 ?

anonymous · Answer

no... $\large 4x^2+2=0 $ has no REAL solution.  So your function does not have any VERTICAL asymptotes.

anonymous · Answer

Vertical asymptotes: 

4x^2 + 2 = 0
Here 4x^2 + 2 has no real roots but does have imaginary roots. But since we are only dealing with real numbers here, we will overlook that. And so 4x^2 + 2 has no real roots which means it is either always negative or always positive, but in this case it's always positve, i.e. 4x^2 + 2 >= 0 for all values of x.

From this we can say that the rational function has no vertical asymptotes because the denominator is never equal to 0.

Horizontal asymptotes:

Since the degree of the denominator is higher than that of the numerator, the horizontal asymptote must be y = 0

anonymous · Answer

Thank you so much! I was CLOSE! I am taking this calc course online and sometimes reading the book by myself is very challenging! I appreciate the help!