How to Find Horizontal Asymptotes
A horizontal asymptote is a horizontal line that the graph of a function approaches as the input approaches positive or negative infinity. In other words, it is a line that the graph gets closer and closer to, but never actually touches.
To find the horizontal asymptotes of a function, we need to look at the behavior of the function as the input approaches infinity. There are two cases to consider:
- Case 1: The limit of the function as the input approaches infinity exists and is finite. In this case, the horizontal asymptote is the line $y = L$, where $L$ is the limit.
- Case 2: The limit of the function as the input approaches infinity does not exist or is infinite. In this case, the function does not have a horizontal asymptote.
Example 1
Find the horizontal asymptotes of the function $f(x) = \frac{x^2 + 1}{x-1}$.
Solution:
First, we need to find the limit of the function as $x$ approaches infinity. We can do this by dividing both the numerator and the denominator by $x$:
$$\lim{x\to\infty} \frac{x^2 + 1}{x-1} = \lim{x\to\infty} \frac{\frac{x^2}{x} + \frac{1}{x}}{\frac{x}{x} – \frac{1}{x}} = \lim{x\to\infty} \frac{x + \frac{1}{x}}{1 – \frac{1}{x}} = \lim{x\to\infty} \frac{x}{1} = \infty$$
Since the limit does not exist, the function does not have a horizontal asymptote.
Example 2
Find the horizontal asymptotes of the function $f(x) = \frac{x^2 – 1}{x^2 + 1}$.
Solution:
First, we need to find the limit of the function as $x$ approaches infinity. We can do this by dividing both the numerator and the denominator by $x^2$:
$$\lim{x\to\infty} \frac{x^2 – 1}{x^2 + 1} = \lim{x\to\infty} \frac{\frac{x^2}{x^2} – \frac{1}{x^2}}{\frac{x^2}{x^2} + \frac{1}{x^2}} = \lim_{x\to\infty} \frac{1 – \frac{1}{x^2}}{1 + \frac{1}{x^2}} = \frac{1}{1} = 1$$
Since the limit is finite, the horizontal asymptote is the line $y = 1$.
FAQ
Q: What is the difference between a horizontal asymptote and a vertical asymptote?
A: A horizontal asymptote is a horizontal line that the graph of a function approaches as the input approaches infinity. A vertical asymptote is a vertical line that the graph of a function approaches as the input approaches a specific value.
Q: How do I know if a function has a horizontal asymptote?
A: To determine if a function has a horizontal asymptote, you need to find the limit of the function as the input approaches infinity. If the limit exists and is finite, then the function has a horizontal asymptote. If the limit does not exist or is infinite, then the function does not have a horizontal asymptote.
Q: How do I find the equation of a horizontal asymptote?
A: If the limit of the function as the input approaches infinity exists and is finite, then the equation of the horizontal asymptote is $y = L$, where $L$ is the limit.
Q: Can a function have more than one horizontal asymptote?
A: No, a function can only have one horizontal asymptote.
