How do you find the asymptotes for #(t^3 - t^2 - 4t + 4) /(t^2 + t - 2)#?

Question

Theodore Amidon · Accepted Answer

There isn't an asymptote.

The way asymptotes work is that there is a value that #x#, or in this case #t#, cannot equal when in the denominator, or else it would be dividing by zero. What this looks like when graphed is a value that the line will get closer and closer to but never touch. The way to find an asymptote is to factor everything, cross out what divides out, and then solve for #t# in the denominator. So first, we factor. #t^2+t-2# can be factored to #(t-1)(t+2)#. The top one is a little harder, but through synthetic division I found that #(x-1)# is a factor, which leaves #(t^2-4)#, which we can simplify with the "difference of squares" to become #(t-2)(t+2)#. All together we now have #((t-1)(t+2)(t-2))/((t-1)(t+2))#. Now we move on to the second step, dividing out factors that are the same in the numerator as the denominator. #(cancel(t-1)cancel(t+2)(t-2))/(cancel(t-1)cancel(t+2))#. We are left with #(t-2)/1#, or just #t-2#. Now, remember that an asymptote is the value that #x# cannot equal or else it would be dividing by zero. But in this case, #x# has no such constraints because #x# isn't even in the denominator. So, there is no asymptote as it is now a linear equation. If you don't believe me, let's graph our original equation. graph{y=((x-1)(x^2-4))/(x^2+x-2)} There is clearly no asymptote, and we can see that from the graph and from the algebra we did. Nice job, thanks for sticking with me through this problem. Good work!

Elijah Allen · Answer

undefined To find the asymptotes for the rational function $ \frac{t^3 - t^2 - 4t + 4}{t^2 + t - 2} $, follow these steps:

1. Check for any vertical asymptotes by finding the values of $ t $ that make the denominator zero. These values represent vertical asymptotes if they are not canceled out by the numerator.
   
2. Factor the denominator $ t^2 + t - 2 $ to find its roots, which will be the points where vertical asymptotes may occur.

3. Check for any horizontal asymptotes by examining the behavior of the function as $ t $ approaches positive or negative infinity.

4. If the degrees of the numerator and denominator polynomials are equal, divide the leading coefficient of the numerator by the leading coefficient of the denominator to determine the horizontal asymptote equation.

5. If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

6. If the degree of the denominator is greater than the degree of the numerator, the horizontal asymptote is $ y = 0 $.

Following these steps will help you identify any vertical or horizontal asymptotes of the given rational function.