Horizontal Asymptote at y=3 and Vertical Asymptote at x = 4
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When a function has a horizontal asymptote at y=3, it means that as x approaches positive or negative infinity, the function gets closer and closer to the horizontal line y=3. This can be written as:
lim (x→±∞) f(x) = 3
When a function has a vertical asymptote at x=4, it means that as x approaches 4 from either side, the function approaches positive or negative infinity. This can be written as:
lim (x→4) f(x) = ±∞
To find a function that has both a horizontal asymptote at y=3 and a vertical asymptote at x=4, we can use a rational function of the form:
f(x) = (ax + b) / (x – 4)
where a and b are constants to be determined.
To have a horizontal asymptote at y=3, the numerator and denominator of the function must have the same degree. Therefore, we will set a=1 and b=3, so that:
f(x) = (x + 3) / (x – 4)
This function has a horizontal asymptote at y=3, because as x approaches infinity or negative infinity, the highest degree term in the numerator (x) will dominate, and the highest degree term in the denominator (x) will also dominate, so the function will approach y=3.
This function also has a vertical asymptote at x=4, because as x approaches 4 from either side, the denominator (x-4) will approach zero, causing the function to approach positive or negative infinity, depending on the sign of the numerator (x+3).
Therefore, the function f(x) = (x + 3) / (x – 4) has a horizontal asymptote at y=3 and a vertical asymptote at x=4.
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