Limits at Infinity: Understanding Concepts and Calculation Strategies

Limit is Infinity (Infinite Limit): This refers to the situation where the y-values of a function approach positive or negative infinity as x approaches a finite number (e.g., \lim_{x \to 2} f(x) = \infty). This is typically associated with vertical asymptotes. The infinity indicates vertical movement (up/down).
Limit at Infinity: This refers to the situation where we are examining the behavior of a function as x approaches positive or negative infinity (e.g., \lim{x \to \infty} f(x) = L or \lim{x \to -\infty} f(x) = L). This is associated with horizontal asymptotes and end behavior. The infinity indicates horizontal movement (left/right).

x\to\infty: Means we are moving to the right indefinitely along the x-axis.
x\to\-\infty: Means we are moving to the left indefinitely along the x-axis.
Interpreting Results:
- If the answer (the limit's value) is positive infinity, it means y goes up.
- If the answer is negative infinity, it means y goes down.
- Combining: The plus/minus inside the limit (x\to\pm\infty) refers to left/right on the x-axis. The plus/minus outside the limit (the resulting \pm\infty) refers to up/down on the y-axis.
End Behaviors: Limits at infinity describe the function's behavior at the