# How do you identify all asymptotes or holes and intercepts for #f(x)=(x+3)/(x^2+7x+12)#?

##### 1 Answer

First you need to factor the denominator.

Now you will have

Now cancel out any like terms, in this case the

Now

to find your y component, plug in

so

This equals to

so the coordinate for your hole would be

Now you still have the

- note that when two like values cancel out there is a whole and when there are no values cancelling out then the zero of
#x# in the denominator will be your V.A. *

Now we can check for horizontal asymptotes. H.A.s exist only when the numerator's leading degree is less than or equal to the leading degree of the denominator.

In this case the degree of the numerator is x and the degree of the denominator is

Finally we can check for Slant asymptotes. S.A.s exist where the degree of the numerator is 1 greater than the degree of the denominator. In this case the numerator has a degree smaller than the denominator. This means we will not have a slant asymptote.

We look at the zeros of the numerator however we already cancelled out the zero

To find the y intercept plug in 0 for x in the remaining equation which is

SUMMARY

Hole at

V.A, at

H.A. at

S.A: does not exist

X int: does not exist

Y int: