Limit of Functions at Infinity Quiz

The concept of limx→∞ f(x) = L means that as the input variable x approaches infinity, the output of the function f(x) approaches the limit L. This can be expressed as ∀ε∃N∀x[(x > N) → (|f(x) − L| < ε)], indicating that for any positive value ε, there exists a value N such that for all x greater than N, the difference between f(x) and L is less than ε.

For limx→∞ f(x) to not exist, it means that there is no finite limit L that f(x) approaches as x tends to infinity. This can be expressed as ∃ε∀N∃x[(x > N) ∧ (|f(x) − L| ≥ ε)], indicating that there exists a positive value ε such that for all N, there exists an x greater than N for which the difference between f(x) and any potential limit L is greater than or equal to ε.

The quantifiers (∀, ∃) and relations (<, ≥) in the expressions of limx→∞ f(x) = L and limx→∞ f(x) does not exist play a crucial role in defining the behavior of the function f(x) as x approaches infinity. The quantifiers indicate universal (∀) or existential (∃) quantification, while the relations (<, ≥) specify the nature of the relationship between the function's output and the limit L. These elements are essential for precisely describing the behavior of real-valued functions as the input variable grows without bound.