Understanding Function Behavior as x Approaches Infinity and Negative Infinity: Explained with Examples

as x→∞, f(x) → ∞ as x→-∞, f(x) → -∞

In mathematics, as x approaches positive infinity (x → ∞), a function f(x) is said to approach positive infinity (f(x) → ∞) if the values of f(x) become arbitrarily large as x becomes larger and larger

In mathematics, as x approaches positive infinity (x → ∞), a function f(x) is said to approach positive infinity (f(x) → ∞) if the values of f(x) become arbitrarily large as x becomes larger and larger.

Similarly, as x approaches negative infinity (x → -∞), a function f(x) is said to approach negative infinity (f(x) → -∞) if the values of f(x) become arbitrarily large in the negative direction as x becomes more and more negative.

To better understand this concept, let’s look at some examples.

Example 1:
Consider the function f(x) = x^2. As x becomes larger and larger (x → ∞), the values of f(x) = x^2 also become larger and larger, approaching positive infinity (f(x) → ∞). Similarly, as x becomes more and more negative (x → -∞), the values of f(x) = x^2 become larger and larger in the positive direction, approaching positive infinity as well.

Example 2:
Now consider the function f(x) = -2x + 1. As x becomes larger (x → ∞), the values of f(x) = -2x + 1 become more and more negative, approaching negative infinity (f(x) → -∞). Similarly, as x becomes more and more negative (x → -∞), the values of f(x) = -2x + 1 become larger and larger in the positive direction, approaching positive infinity.

In both examples, as x approaches infinity or negative infinity, the function values approach the respective infinities stated. This behavior is often used to describe the overall trend or behavior of a function as it approaches extreme values of x.

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