Use limits to describe the behavior of the rational function near the indicated asymptote
f(x)=3/x-4
To describe the behavior of the rational function f(x) = 3/(x - 4) near the indicated asymptote, we need to evaluate the limits as x approaches the asymptote.
In this case, the given function has a vertical asymptote at x = 4. To describe the behavior near the asymptote, we will evaluate the limit as x approaches 4 from both the left and the right.
1. Left-hand limit (x → 4-):
To evaluate the left-hand limit, we substitute values of x that are slightly less than 4 into the function.
lim(x → 4-) 3/(x - 4) = ∞
As x approaches 4 from the left side, the function tends towards positive infinity (∞).
2. Right-hand limit (x → 4+):
To evaluate the right-hand limit, we substitute values of x that are slightly greater than 4 into the function.
lim(x → 4+) 3/(x - 4) = -∞
As x approaches 4 from the right side, the function tends towards negative infinity (-∞).
Therefore, the behavior of the rational function near the vertical asymptote x = 4 is that it approaches positive infinity (∞) as x approaches 4 from the left side and approaches negative infinity (-∞) as x approaches 4 from the right side.