Find the Asymptotes f(x)=(2x^2+1)/x

Problem

ƒ(x)=(2*x2+1)/x

Solution

1. Identify vertical asymptotes by finding the 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*(2*x2+1)>degree(x)

3. Find the slant (oblique) asymptote by performing polynomial long division or dividing each term in the numerator by the denominator.

ƒ(x)=(2*x2)/x+1/x

4. Simplify the expression to identify the linear part of the function.

ƒ(x)=2*x+1/x

5. Determine the slant asymptote by taking the limit as x approaches infinity, where the remainder term 1/x approaches zero.

y=2*x

Final Answer

Vertical Asymptote: *x=0, Slant Asymptote: *y=2*x

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