SOLUTION: hey i need help with this problem...... For each rational function, identify any holes or horizontal or vertical asymptotes of its graph. y= x+5/(x-2)(x-3) Thanks alot!!

Question 136117: hey i need help with this problem......
For each rational function, identify any holes or horizontal or vertical asymptotes of its graph.
y= x+5/(x-2)(x-3)
Thanks alot!!
Found 2 solutions by checkley77, JSmall:
Answer by checkley77(12844) (Show Source): You can put this solution on YOUR website!
y= x+5/(x-2)(x-3)
y=(x+5)/(x^2-5x+6)
(graph 300x200 pixels, x from -6 to 5, y from -10 to 20, (x+5)/(x^2-5x+6) ).

Answer by JSmall(7) (Show Source): You can put this solution on YOUR website!
y= x+5/(x-2)(x-3)
"Holes" would occur when the numerator and denominator have a common factor which is an expression which could be zero. Since this fraction has no such common factors, there will be no "holes".
Vertical asymptotes will occur for x-values which make the denominator zero. For this equation the denominator will be zero if x = 2 or x = 3. So the vertical lines, x = 2 and x = 3 will be vertical asymptotes for this equation.
Horizontal asymptotes will occur if y approaches some constant value when x-values become very large (positive or negative).
For this equation, when x-values become very large, the denominator of the fraction becomes very large. This makes the fraction very, very small. In fact the fraction approaches zero in value. So as x-values become very large, the fraction becomes negligible and the y-value approaches the value of the non-fraction portion of the right side: x.
Since x is not a constant value there will be no horizontal asymptotes. But there will be what is called a slant asymptote! The line y = x will be a slant asymptote.

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