Rational Functions 
 Section 5.6  
Find the vertical and horizontal asymptotes, if any. 
Solve 
Step 

VA: Set the denominator equal to zero. 
Set x  3 = 0 So, x = 3.  Obviously, f(3) = undefined. What happens as x approaches 3? i.e. x → 3 
From the left hand side of 3 
From the
right hand side of 3  
The Vertical Asymptote, VA, is the line, x = 3. 


From the sketch of the graph we see y = 0 is the horizontal asymptote. Look what happens when we plug in large and small numbers. 

The Horizontal Asymptote, HA, is the line, y = 0 
Whenever the degree of the numerator is less than the degree of the denominator, the HA is y = 0. 

As x →∞ (or ∞) , f (x) → 0 because the denominator will increase much faster than the numerator. 


Find the Vertical Asymptotes, if any. 
[Solution] 


Find the vertical and horizontal asymptotes, if any.
Then find x and y intercepts, if any. 
Solve 
Step 

Vertical Asymptote: x  2 = 0, x = 2 
The vertical asymptote is the line x = 2  
Horizontal Asymptote: y = 3 
The horizontal asymptote is the line y = 3  
If the degree of the numerator and the denominator are the same, as x approaches ±∞ , then y approaches the coefficients of the of the leading term in the numerator and denominator. 

x  intercept: (4/3, 0) 
Set y = g(x) = 0.  The only way a fraction can equal zero is if the numerator is zero. 


y  intercept: (0, 2) 
g(0) = 2  Set x = 0. 
Let's put it all together in a graph. 

Notice the x and y intercepts are denoted by the blue dots. The asymptotes are denoted by the broken blue lines. 


Find the vertical and horizontal asymptotes, if any. Then find x
and y intercepts, if any. 
[Solution] 


Is there really an Asymptote? 
Solve 
Step 
Property 
Vertical Asymptote: None 

VA: Set the denominator equal to zero.  Is this really an asymptote??  
Something
is wrong. As x →4,
f→6. This is not
an asymptote but a hole in the After we
reduce the fraction, we get 

Horizontal Asymptote: None 
as x → ∞, f (x) = x + 2 → ∞  as x → ∞, f (x) = x + 2 → ∞ 
Since the factor (x  4) cancels out, we are left with the line y = x + 2 where x ≠ 4.  

Is there really an Asymptote? 
[Solution] 


Tutorials and Applets by
Joe McDonald
Community College of Southern Nevada