Horizontal asymptotes
lim as x→ ±∞ equals a constant
Horizontal asymptotes are lines on a graph that a curve approaches but does not touch as the curve extends infinitely in the positive or negative direction. They are important in calculus because they can help us understand the long-term behavior of a function.
To find the horizontal asymptote of a function, we must first determine if the function approaches a finite value as x approaches positive or negative infinity, or if it grows without bound. We do this by taking the limit of the function as x goes to infinity or negative infinity.
If the limit exists and is finite, then the horizontal asymptote is the value of that limit. If the limit does not exist or is infinite, then there is no horizontal asymptote.
For example, consider the function f(x) = (x^2 + 3)/(2x^2 – 5x + 1). To find the horizontal asymptote, we take the limit of the function as x approaches infinity:
lim x->∞ (x^2 + 3)/(2x^2 – 5x + 1) = lim x->∞ (1 + 3/x^2)/(2 – 5/x + 1/x^2) = 1/2
Similarly, taking the limit of f(x) as x approaches negative infinity, we get:
lim x->-∞ (x^2 + 3)/(2x^2 – 5x + 1) = lim x->-∞ (1 + 3/x^2)/(2 – 5/x + 1/x^2) = 1/2
Therefore, the horizontal asymptote of f(x) is y = 1/2.
We can also have vertical asymptotes, a vertical asymptote is a vertical line on the graph of a function that the graph approaches but does not touch as the input of the function approaches a certain value. To find the vertical asymptotes, we must find where the denominator of the function equals zero. Then, we must determine if any of these values make the numerator go to zero as well. If none of the values make the numerator go to zero, then we have a vertical asymptote at that value of x.
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