A curve asymptote is a line where the distance between the curve and the line approaches 0. The function is undefined for the value of x=(5/2). Thus, x=(5/2) is an asymptote.
<h3>What are asymptotes?</h3>
A curve asymptote is a line where the distance between the curve and the line approaches 0 when one or both of the x or y coordinates approaches infinity.
The asymptotes are the values for which the function is not defined. The asymptotes of a fractional function are found by equating its denominator's factors against zero. Therefore, the value of the asymptotes is,


Now, substitute the value of x as (5/2) and 5, to know if the function is defined or not.
}{[2(5)-5](5-5)}\\\\\\f(5) = \dfrac{(35-1)(0)}{(10-5)(0)} = 0](https://tex.z-dn.net/?f=f%28x%29%20%3D%20%5Cdfrac%7B%287x-1%29%28x-5%29%7D%7B%282x-5%29%28x-5%29%7D%5C%5C%5C%5C%5C%5Cf%285%29%20%3D%20%5Cdfrac%7B%5B7%285%29-1%5D%285-5%29%7D%7B%5B2%285%29-5%5D%285-5%29%7D%5C%5C%5C%5C%5C%5Cf%285%29%20%3D%20%5Cdfrac%7B%2835-1%29%280%29%7D%7B%2810-5%29%280%29%7D%20%3D%200)
Since for the value of x=5, the function is defined and returns the value as 0. Thus, x=5 is not an asymptote.
}{[2(\frac52)-5](\frac52-5)}\\\\\\f(\frac52) = \dfrac{(16.5)(-2.5)}{(0)(-2.5)} = \dfrac{\infty}{0}](https://tex.z-dn.net/?f=f%28x%29%20%3D%20%5Cdfrac%7B%287x-1%29%28x-5%29%7D%7B%282x-5%29%28x-5%29%7D%5C%5C%5C%5C%5C%5Cf%28%5Cfrac52%29%20%3D%20%5Cdfrac%7B%5B7%28%5Cfrac52%29-1%5D%28%5Cfrac52-5%29%7D%7B%5B2%28%5Cfrac52%29-5%5D%28%5Cfrac52-5%29%7D%5C%5C%5C%5C%5C%5Cf%28%5Cfrac52%29%20%3D%20%5Cdfrac%7B%2816.5%29%28-2.5%29%7D%7B%280%29%28-2.5%29%7D%20%3D%20%5Cdfrac%7B%5Cinfty%7D%7B0%7D)
Since the function is undefined for the value of x=(5/2). Thus, x=(5/2) is an asymptote.
Learn more about Asymptotes:
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