Find the Asymptotes f(x)=(x^2-25)/x

Problem

ƒ(x)=(x2−25)/x

Solution

1. Identify vertical asymptotes by finding values of x where the denominator is zero and the numerator is non-zero.

x=0

2. Check for horizontal asymptotes by comparing the degrees of the numerator and denominator. Since the degree of the numerator (2) is greater than the degree of the denominator (1), there is no horizontal asymptote.

degree*(x2−25)>degree(x)

3. Find the slant (oblique) asymptote by performing polynomial division or simplifying the fraction.

ƒ(x)=(x2)/x−25/x

4. Simplify the expression to determine the linear part of the function as x approaches infinity.

ƒ(x)=x−25/x

5. Determine the slant asymptote equation by taking the linear part of the result, as the term 25/x approaches zero.

y=x

Final Answer

Vertical Asymptote: *x=0, Slant Asymptote: *y=x

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