Question

Consider the function f(x) = the quantity x squared minus 25 end quantity over x plus 4. Determine all asymptotes of this function including horizontal, vertical, and oblique (slant).

Answer 1

@KinzaN

Answer 2

Do you know when you have an oblique asymptote in rational functions?

Answer 3

^^^

Answer 4

When the highest degree of the numerator is one more than the highest degree of the denominator. In simpler terms, it's when the biggest exponent on top is one more than the biggest exponent on the bottom, like in this case.

Answer 5

Since you cannot cancel any factors after factoring the numerator, which is a difference of squares, there are no point discontinuities--no holes!

Answer 6

So would be the answer? how would I anser all of the requirements?

Answer 7

Well, there is a vertical asymptote. The denominator cannot be 0, so the x value that would make the denominator equal to 0 cannot exist.

Answer 8

There is a vertical asymptote when x = -4.

Answer 9

To find the oblique asymptote algebraically requires a lot of explanation. But the short-cut is to perform the division. Your quotient will be the equation of the oblique asymptote.

Answer 10

So that would be (x^2 - 25)/(x+4)

Answer 11

Both synthetic and long division will solve the problem.

Answer 12

So what would it be?

Answer 13

and what is the horizontal one?

Answer 14

You cannot have both a horizontal and an oblique asymptote.

Answer 15

It's either one or the other.

Answer 16

But you CAN have both a vertical asymptote and a point discontinuity. It's confusing.

Answer 17

So whats my oblique then?

Answer 18

so far my answer would be
There is a vertical asymptote when x = -4. There is no horizontal asymptote and the oblique is .....

Answer 19

@KinzaN hello?

Answer 20

Can you do the division? You must have learned it, lol

Answer 21

Divide the numerator by the denom. and your quotient is the equation of the oblique asymptote.

Answer 22

I get x-4-9/x+4. @KinzaN is that correct?

Answer 23

I'll check for you. :)

Answer 24

Ok thank you

Answer 25

Ah, you don't add the remainder. The quotient is simply x - 4, so the equation of the oblique slant is y = x - 4

Answer 26

oh.. ok thanks