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- When considering functions made up of the sums, differences, products or quotients of different sorts of functions (polynomials, exponentials and logarithms), or different powers of the same sort of function we say that one function dominates the other.
teachingcalculus.com/2012/08/08/158/
Feb 8, 2019 · The distinction between $x \to \infty$ and $x \to 0$ that you seem to remember might be that for $x \to \infty$, higher powers $x^k$ dominate over lower powers, but as $x \to 0$ it's the other way around.
When one function dominates another, then it approaches infinity at a faster level than the other function. Since the dominant function approaches faster and it is in the denominator, then it drives the quotient to . Our initial order of dominance looks like this.
One set is said to dominate another if there is a function from the latter into the former. More formally, we have the following. Definition: Dominance. If A and B are sets, we say “ A dominates B ” and write | A |> | B | iff there is an injective function f with domain B and codomain A.
Consider the functions \( x^{\text{sin}(x)} \) and \( e^{\text{cos}(x)} \). As \( x \) approaches infinity, they oscillate and do not have a limit that would make one dominate the other. Therefore, neither function dominates the other.
- Matt Boelkins
We say that one function dominates another if the magnitude of the ratio of the first function to the second increases without bound as the input increases without bound. More concisely, f(x) dominates g(x) if limx→∞g(x)f(x)=0.
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First, let’s recall what it means for one function to dominate another function. For eventually positive functions 𝑓 and 𝑔, if the limit as 𝑥 approaches ∞ of 𝑓 of 𝑥 divided by 𝑔 of 𝑥 is equal to ∞, then we say that 𝑓 of 𝑥 dominates 𝑔 of 𝑥.
