# How do you find the vertical, horizontal or slant asymptotes for #f(x) = (2x^2 + x + 2) /( x + 1)#?

Slant Asymptote:

Vertical Asymptote:

Horizontal Asymptote: None

The given function is

To find the slant asymptote, divide numerator by the denominator of the given rational function.

The result of the division is

The whole number part of the quotient which is

which is the

To solve for the Vertical Asymptote, use the divisor and equate to zero

and the Vertical Asymptote is

There is

Kindly see the graph of the function

God bless....I hope the explanation is useful.

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Vertical asymptote at

No horizontal asymptote

Slant asymptote:

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To find the vertical asymptote(s), set the denominator equal to zero and solve for ( x ). In this case, set ( x + 1 = 0 ), so ( x = -1 ). Thus, there's a vertical asymptote at ( x = -1 ).

To find the horizontal asymptote, compare the degrees of the numerator and the denominator. Since the degree of the numerator (2) is less than the degree of the denominator (1), there is no horizontal asymptote.

To check for slant asymptotes, perform polynomial long division of the numerator by the denominator. After dividing ( (2x^2 + x + 2) ) by ( (x + 1) ), if there's a remainder, there is a slant asymptote. If the remainder is not zero, it indicates a slant asymptote. If the remainder is zero, there is no slant asymptote.

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