Simplest problem for finding Asymptote.

Algebra Level 1

( x − 5 ) 2 ( x − 5 ) ( x − 3 ) \frac{(x-5)^2}{(x-5)(x-3)}

What are the vertical and horizontal asymptotes?

y = 1 , x = 3 y=1, \quad x=3 y = 1 , x = 5 y=1, \quad x=5 y = − 1 , x = 3 y=-1, \quad x=3 y = − 1 , x = 5 y=-1, \quad x=5

Yuliya Skripchenko by Yuliya Skripchenko , 38, USA

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1 solution

Rafael Dubois
Jul 5, 2018

Before going for the asymptotes, we must simplify the function. We have (x-5)² on the top with (x-5) dividing, so we can cancel them out. We need to do this in order to execute the next step correctly. We are left with (x-5)/(x-3), and for the vertical asymptotes, we must find which value of x can make this function tend to infinity. We know that a limit with denominator 0 tends to infinity, which is why we want the denominator to tend to 0. Then, we do x-3=0, and we get that the function has a vertical asymptote at x=3. Then, we do the limit as x goes to infinity of the whole function. As x gets larger in (x-5)/(x-3), the numerator and denominator become more and more close to each other, so the limit is 1. This means that we have an horizontal asymptote at y=1.

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