|
Question 125252:
5) Find the equations of the horizontal and vertical asymptotes for the following. Type none if the function does not have an asymptote.
a) f (x)= 2x+3/x+2
Answer:
Horizontal:
Vertical:
b) g (x) = 5x/x^+1
Answer:
Horizontal:
Vertical:
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! a)
 Start with the given function
Looking at the numerator  , we can see that the degree is  since the highest exponent of the numerator is . For the denominator  , we can see that the degree is since the highest exponent of the denominator is .
Horizontal Asymptote:
Since the degree of the numerator and the denominator are the same, we can find the horizontal asymptote using this procedure:
To find the horizontal aysmptote, first we need to find the leading coefficients of the numerator and the denominator.
Looking at the numerator , the leading coefficient is 
Looking at the denominator , the leading coefficient is
So the horizontal aysmptote is the ratio of the leading coefficients. In other words, simply divide by to get 
So the horizontal asymptote is 
--------------------------------------------------
Vertical Asymptote:
To find the vertical aysmptote, just set the denominator equal to zero and solve for x
 Set the denominator equal to zero
 Subtract 2 from both sides
 Combine like terms on the right side
So the vertical asymptote is
Notice if we graph  , we can visually verify our answers:
 Graph of with the horizontal asymptote (blue line) and the vertical asymptote (green line)
b)
 Start with the given function
Looking at the numerator  , we can see that the degree is since the highest exponent of the numerator is . For the denominator  , we can see that the degree is since the highest exponent of the denominator is .
Horizontal Asymptote:
Since the degree of the numerator (which is ) is less than the degree of the denominator (which is ), the horizontal asymptote is always 
So the horizontal asymptote is
--------------------------------------------------
Vertical Asymptote:
To find the vertical aysmptote, just set the denominator equal to zero and solve for x
 Set the denominator equal to zero
 Subtract 1 from both sides
 Combine like terms on the right side
 Take the square root of both sides
Since you cannot take the square root of a negative number, the answer is not a real number. So in this case, there are no vertical asymptotes.
Notice if we graph  , we can visually verify our answers:
 Graph of with the horizontal asymptote (blue line) . Notice how there are no vertical asymptotes
|
|
|
| |
