The key difference lies in what is approaching infinity: infinite limits deal with the function's value approaching infinity, while limits at infinity deal with the variable approaching infinity.
Here's a breakdown:
Infinite Limits
Infinite limits occur when the value of a function, f(x), grows without bound (approaches positive or negative infinity) as x approaches a specific value, a. Mathematically, we write:
- limx→a f(x) = ∞
- limx→a f(x) = -∞
Example:
Consider the function f(x) = 1/x2. As x approaches 0, f(x) approaches infinity. Therefore:
limx→0 (1/x2) = ∞
In this case, the limit itself is infinite. The x value approaches a finite number (0).
Limits at Infinity
Limits at infinity occur when we examine the behavior of a function, f(x), as the independent variable, x, grows without bound (approaches positive or negative infinity). Mathematically, we write:
- limx→∞ f(x) = L
- limx→-∞ f(x) = L
Where L is a finite number. It's also possible that these limits are infinite, meaning f(x) grows without bound as x grows without bound.
Example:
Consider the function f(x) = (2x - 1) / x. As x approaches infinity, f(x) approaches 2. Therefore:
limx→∞ (2x - 1) / x = 2
In this case, x itself is approaching infinity. The limit, however, is a finite number (2).
Summary Table
| Feature | Infinite Limits | Limits at Infinity |
|---|---|---|
| Focus | Function value approaching infinity | Variable approaching infinity |
| Notation | limx→a f(x) = ∞ (or -∞) | limx→∞ f(x) = L (L can be ∞) |
| Variable (x) | Approaches a finite value (a) | Approaches infinity (∞ or -∞) |
| Function (f(x)) | Approaches infinity (∞ or -∞) | Approaches a finite value (L) or infinity |
In essence, infinite limits tell us what happens to a function when we get extremely close to a particular x value, while limits at infinity tell us what happens to the function's value as x becomes extremely large (positive or negative).
