Consider the following function f x 3 x2 cos x x2 9 a find...

Name: consider the following function f x 3 x2 cos x x2 9 a find all vertical asymptotes of f x and state any type of discontinuity that you determine in the process b examine the long term behavi 40546
Uploaded: 2020-06-09T21:22:45-08:00
Duration: 4 min 46 s
Channel: Cinsy Krehbiel
Description: consider the following function f x 3 x2 cos x x2 9 a find all vertical asymptotes of f x and state any type of discontinuity that you determine in the process b examine the long term behavi 40546

Consider the following function f (x) = (3 − x)(2 + cos (x)) x2 − 9 . (a) Find all vertical asymptotes of f (x) and state any type of discontinuity that you determine in the process. (b) Examine the long term behaviour of f (x), i.e., as x -> \pm \infty and state any horizontal asymptotes

Transcript

00:01 We're trying to determine what does this function do? what are its asymptotes? how does it behave at the extremes? so to do that, we're going to reduce it to its simplest form.

00:13 That means we're going to want to factor the numerator.

00:16 It's the difference between perfect square, so it's going to come out to be x minus 3 times x plus 3.

00:23 And our denominator is already factored.

00:28 Now we can cancel the common factors.

00:30 That's how we reduce it.

00:31 So this x minus three can cancel.

00:34 We do have to remember that three is excluded from the domain.

00:40 If we were to substitute three into this function, we'd get undefined.

00:44 So we have to remember that.

00:45 But now our function, except for that little discontinuity, will behave the exact same as this, which is x plus three over x.

00:56 In order to manage this and figure out what the limits are, we're going to separate the fraction.

01:01 So we're going to take x over x, which is one, plus three over x.

01:10 Now we can take some limits.

01:12 So what happens as x approaches infinity? 1 over x approaches zero.

01:22 And if it's positive infinity, we're going to approach zero from the positive.

01:26 As x approaches negative infinity, 1 over x approaches 0 approaches 0, but from the negative.

01:38 So either way it approaches zero, whether we're approaching positive or negative infinity with our x, and that means the fraction 3 over x is going to make a negligible contribution to the value of our function...

Question

Consider the following function f (x) = (3 − x)(2 + cos (x)) x2 − 9 . (a) Find all vertical asymptotes of f (x) and state any type of discontinuity that you determine in the process. (b) Examine the long term behaviour of f (x), i.e., as x -> \pm \infty and state any horizontal asymptotes

Please give Ace some feedback