If h(x)~g(x), is h(x+1)~g(x)?
I'm not sure that implication holds in general... I can imagine failure can occur if the functions grow sufficiently fast, or if they can do other odd things, like zig-zag back and forth.
I think it might be easier to first decide if pi(x) ~ pi(x + 1), or if (x - 1) / ln (x - 1) ~ x / ln x
An asymptotic function is a mathematical function that describes the behavior of a function as its input approaches a certain value, typically infinity or zero. It is used to analyze the long-term behavior of functions and can help determine the growth rate or decay rate of a function.
Asymptotic functions are different from regular functions in that they do not necessarily describe the exact behavior of a function, but rather its long-term behavior. Regular functions provide specific values for a given input, while asymptotic functions provide a general trend or approximation for a large range of inputs.
The main properties of asymptotic functions include the limit behavior, rate of growth or decay, and big-O notation. These properties can be used to classify and compare different functions based on their long-term behavior.
Asymptotic functions are used in many fields, including physics, engineering, and computer science. They can be used to model physical phenomena, analyze the efficiency of algorithms, and predict the behavior of complex systems.
While asymptotic functions can provide useful insights into the behavior of functions, they are not always accurate for small or medium-sized inputs. Additionally, they may not accurately capture the behavior of functions with oscillating or discontinuous behavior.