Question

What are the equations of asymptotes of the graph of the function f(x)=3x^2-2x-1 / x^2+3x-10

Answer 1

Please use parentheses around numerator and denominator here to reduce the chances of mis-reading your post.  Better yet, write the given function vertically:

          3x^2-2x-1  
 f(x)= ---------------
          x^2+3x-10

horiz. asy.:  Find the limit of 3x^2 / x^2 as x grows large without bound.  It is 3.
Thus, the horiz. asy. is y = 3.

Vert. asymptotes:  Factor the denominator, set the factors = to 0 and solve the resulting equations for x:  x^2 + 3x - 10 = (x+5)(x-2).  So, x=-5 and x=2 are your vertical asymptotes.

Answer 2

An asymptote is of a graph of a function is a line that continually approaches a given curve but does not meet it at any finite distance.
There are three major types of asymptote: Vertical, Horizontal and Oblique asymptotes.
Vertical asymptotes are vertical lines which correspond to the zeroes of the denominator of a rational function. They are the values of x for which a rational function is not defined.
Thus given the rational function:
The vertical asymptotes are the vertical lines corresponding to the values of x for which

Solving the above quadratic equation we have:

Therefore, the vertical asymptotes of the function
are x = 2 and x = -5

The horizontal asymptote of a rational function describes the behaviour of the function as x gets very big.The horizontal asymptote is usually obtained by finding the limit of the rational function as x tends to infinity.
For rational functions with the highest power of the variable of the numerator less than the highest power of the variable of the denominator, the horizontal asymptote is usually given by the equation y = 0.
For rational functions with the highest power of the variable of the numerator equal to the highest power of the variable of the denominator, the horizontal asymptote is usually given by the ratio of the coefficients of the highest power of the variable of the numerator to the coefficient of the highest power of the denominator.
Therefore, the horizontal asymptotes of the function
is