Math  /  Algebra

QuestionGiven the function f(x)=x2+9x14x27xf(x)=\frac{-x^{2}+9 x-14}{x^{2}-7 x}, find any removable discontinuities, and vertical and horizontal asymptotes, if they exist. If there is more than one, list the numbers from least to greatest, separated by a comma.
Removable discontinuity at x=x= \square Vertical Asymptote at x=x= \square Horizontal Asymptote at y=y= \square

Studdy Solution
Identify horizontal asymptotes.
The degrees of the numerator and denominator are both 1 (after canceling), so we compare the leading coefficients.
The leading coefficient of the numerator is 1-1 and the leading coefficient of the denominator is 11.
The horizontal asymptote is given by:
y=11=1 y = \frac{-1}{1} = -1
Horizontal Asymptote at y=1 y = -1 .
Removable discontinuity at x=7 x = 7 .
Vertical Asymptote at x=0 x = 0 .
Horizontal Asymptote at y=1 y = -1 .

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