Analyzing the Limit of a Function as x Approaches Infinity and Negative Infinity: Understanding the Long-term Behavior

as x→∞, f(x) → ∞ as x→-∞, f(x) → -∞

To understand the behavior of a function as x approaches infinity or negative infinity, we need to analyze the limit of the function as x approaches these values

To understand the behavior of a function as x approaches infinity or negative infinity, we need to analyze the limit of the function as x approaches these values.

1. As x approaches infinity (x → ∞):
When x approaches infinity, we evaluate the limit of f(x). If the limit approaches positive infinity, it means that as x gets larger and larger, the function also increases without bound. Mathematically, we can represent this as:

lim (x → ∞) f(x) = ∞

This means that as x approaches infinity, f(x) also approaches infinity.

2. As x approaches negative infinity (x → -∞):
Similar to the previous case, when x approaches negative infinity, we evaluate the limit of f(x). If the limit approaches negative infinity, it means that as x becomes infinitely negative, the function also becomes infinitely negative. Mathematically, we can represent this as:

lim (x → -∞) f(x) = -∞

This means that as x approaches negative infinity, f(x) approaches negative infinity.

It’s important to note that these limits represent the long-term behavior of the function as x approaches infinity or negative infinity. The function may still have fluctuations or smaller patterns between certain intervals, but the overall trend is towards positive infinity as x approaches infinity and towards negative infinity as x approaches negative infinity.

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