Limits at Infinity: Understanding Concepts and Calculation Strategies

Limits: At Infinity vs. Infinite

Clarifying Terminology

• 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., $\lim<em>{x \to \infty} f(x) = L$ or $\lim</em>{x \to -\infty} f(x) = L$). This is associated with horizontal asymptotes and end behavior. The infinity indicates horizontal movement (left/right).

Understanding Limits at Infinity Graphically

• $x\to\infty$: 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