Question
Find the horizontal and vertical asymptotes for the graph of y=2x+4/x-6
Answers
GPT 3.5
To find the horizontal asymptote, we need to determine the behavior of the function as x approaches positive or negative infinity.
As x approaches positive infinity, the numerator (2x + 4) grows without bound, while the denominator (x - 6) also grows without bound. Therefore, the fraction will approach positive infinity.
As x approaches negative infinity, both the numerator and the denominator grow without bound but in opposite directions. Therefore, the fraction will approach negative infinity.
Since the function approaches a different value (positive infinity) as x approaches positive infinity and a different value (negative infinity) as x approaches negative infinity, there is no horizontal asymptote.
To find the vertical asymptote, we set the denominator equal to zero and solve for x:
x - 6 = 0
x = 6
Therefore, there is a vertical asymptote at x = 6.
In summary:
- There is no horizontal asymptote
- There is a vertical asymptote at x = 6
As x approaches positive infinity, the numerator (2x + 4) grows without bound, while the denominator (x - 6) also grows without bound. Therefore, the fraction will approach positive infinity.
As x approaches negative infinity, both the numerator and the denominator grow without bound but in opposite directions. Therefore, the fraction will approach negative infinity.
Since the function approaches a different value (positive infinity) as x approaches positive infinity and a different value (negative infinity) as x approaches negative infinity, there is no horizontal asymptote.
To find the vertical asymptote, we set the denominator equal to zero and solve for x:
x - 6 = 0
x = 6
Therefore, there is a vertical asymptote at x = 6.
In summary:
- There is no horizontal asymptote
- There is a vertical asymptote at x = 6