Horizontal and Vertical Asymptotes

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Question

Horizontal and Vertical Asymptotes

Use limits to determine the equations for all vertical asymptotes.

x² + x ― 6

c. y = ------------------

x² + 2x ― 8

Accepted Answer

Welcome back, everyone. Identify the vertical asymptotes for the function F of X equals 3x2 + 7 x minus 6 divided by X2 + x minus 12. So let's understand that we're given a rational function. If we have a rational function in the form of PX divided by Q of X. The vertical asymptotes exist when our denominator is equal to 0, so the function is undefined, right, because division by 0 is not defined. So what we're going to do to identify the vertical asymptotes is essentially set our denominator equal to 0, but we also want to make sure that the numerator for the same X values that we identify. It is not equal to 0, right, because otherwise we have a hole which is a removable discontinuity and it doesn't correspond to a vertical asymptote. So our denominator in this case is X2 + x minus 12. We can factor it out as x minus 3 multiplied by x + 4. And we are basically setting it equal to 0, and since we have two factors, we can first we set X minus 3 equals 0, which gives us our first solution X equals 3. And now for the second factor X + 4, we're setting it equal to 0. And this gives us X equals -4. So those are the candidates for the vertical asymptotes. In order to verify that those are vertical asymptotes, we want to look at the numerator, which is 3x2 + 7 x minus 6, and we want to evaluate that numerator for every X. If the values are not zero, then we essentially confirm that those are vertical asymptotes. So first of all, let's evaluate our new function. Let's call it G of X, and we're going to evaluate G at 0.3, our first candidate. So we're substituting 3 for every X, which gives us 3 multiplied by 3 squad plus 7 multiplied by 3 minus 6. So 3 cubed 27 plus. 7 multiplied by 3, that's 21 minus 6. We got 48 minus 6, which is 42. It is not equal to 0, right? So we have verified that X equals 3 is a vertical asymptote. And now we're going to consider G at -4, which gives us 3 multiplied by -4 squad plus 7 multiplied by -4 minus 6. Now let's evaluate the result. That's 3 multiplied by 1648 minus 28 minus 6. So 20 minus 6 gives us 14. That's not equal to 0 as well, meaning X equals -4 is also a vertical asymptote. So we essentially have two vertical asymptotes, X equals 3 and X equals -4. Those will be our final answers to this problem. Thank you for watching.

Horizontal and Vertical Asymptotes

Use limits to determine the equations for all vertical asymptotes.

x² + x ― 6
c. y = ------------------
x² + 2x ― 8

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Horizontal and Vertical AsymptotesUse limits to determine the equations for all vertical asymptotes.x² + x ― 6c. y = ------------------x² + 2x ― 8

Watch next

Horizontal and Vertical Asymptotes

Use limits to determine the equations for all vertical asymptotes.

x² + x ― 6
c. y = ------------------
x² + 2x ― 8