SOLUTION: 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

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

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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
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