Question

6) Give all asymptotes

Answer 1

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.

What are asymptotes?

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,

$(2x-5)=0\\\\2x=5\\\\x=\dfrac52$

$x-5=0\\\\x=5$

Now, substitute the value of x as (5/2) and 5, to know if the function is defined or not.

$f(x) = \dfrac{(7x-1)(x-5)}{(2x-5)(x-5)}\\\\\\f(5) = \dfrac{[7(5)-1](5-5)}{[2(5)-5](5-5)}\\\\\\f(5) = \dfrac{(35-1)(0)}{(10-5)(0)} = 0$

Since for the value of x=5, the function is defined and returns the value as 0. Thus, x=5 is not an asymptote.

$f(x) = \dfrac{(7x-1)(x-5)}{(2x-5)(x-5)}\\\\\\f(\frac52) = \dfrac{[7(\frac52)-1](\frac52-5)}{[2(\frac52)-5](\frac52-5)}\\\\\\f(\frac52) = \dfrac{(16.5)(-2.5)}{(0)(-2.5)} = \dfrac{\infty}{0}$

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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