Help finding horizontal and vertical asymptotes?

Question

I have to find the horizontal asymptote:
g(x)= -2x+3 / 3x+1

and the vertical asymptote of:
g(x)= x+3 / x(x-5)

Any help is appreciated!

Answer 1

for the horizontal asymptote:

rule #1:
if the highest exponent of a function in the numerator is less than the highest exponent of a function in the denominator, then the horizontal asymptote is y = 0

rule # 2
if the highest exponent of a function in the numerator is greater than the highest exponent of a function in the denominator, then there is NO horizontal asymptote

if the highest exponent of a function in the numerator is equal to the highest exponent of a function in the numerator, then the horizontal asymptote is the ratio of the leading coefficient of the function on the numerator to the leading coefficient of the function in the denominator.

f(x) = (ax + b) / (cx + d)
y = a/c


g(x) = (-2x + 3) / (3x + 1)

use rule number 3
the horizontal asymptote is y = -2/3

g(x) = (x + 3) / (x (x - 5) )

simplify
g(x) = (x + 3) / (x^2 - 5)

use rule number 1
the horizontal asymptote is y = 0


the vertical asymptote is the value(s) of x that makes the function undefined

since you can't divide by a 0, set the denominator equal to 0

3x + 1 = 0
3x = -1
x = -1/3


same thing with the second problem
x (x - 5) = 0
x= 0 or 5

Answer 2

Let's reason through this.

Horizontal asymptote is where Y value will be, if X value becomes extremely large or small. (positive or negative). If you have a function (equation) and make the x value very large, what value would they approach? If the formula is (-2x+3)/(3x+1) what happens?

The answer is the ratio of the leading co-efficient, which is -2/3. You can graph this and verify. Basically, the x value gets so large, constant part becomes irrelavant.

Vertical asymptote is basically asking, for what X value would Y be indeterminate? (unable to determine)

What value can you pick to do this? Can you divide by zero? What value of X would make the denominator zero? You can graph to check this.

Answer 3

horizontal asymptote means the function cannot equal a specific value.
the first one, g(x) cannot equal -2/3, so y = -2/3 is a hor. asym.

vertical asymptotes occur if a function isn't defined somewhere.
x = 0 and x = 5 are hor. asymptotes since x and x - 5 are in the denominator and thus can't equal 0.

Note: snowflake is incorrect. Graph the function and you will see that it is y = -2/3 that is the horizontal asymptote.

Answer 4

The vertical asymptote can be determined by the denominator. Any number that makes the denominator equal zero will be a vertical asymptote. In this case, the vertical asymptotes are x = 0 and x = 5

The horizontal asymptote can be determined by the numerator. Any number that makes the numerator equal zero will be a horizontal asymptote. In this case, y = 3/2 is the horizontal asymptote.