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The concept of a limit is a fundamental part of calculus and analysis, and it's essential in understanding the behavior of functions as they approach a certain point. In the context of slowly varying functions, we use the limit definition to investigate how the ratio of the function at two different points approaches a constant as one point tends to infinity.

The limit of a function as the variable approaches a value or infinity is the value that the function tends to as the variable gets arbitrarily close to that value. Mathematically, we say that the limit of a function f(x) as x approaches a value a is L if for every number ε > 0, there exists a number δ > 0 such that if 0 < |x - a| < δ, then |f(x) - L| < ε. For functions tending to infinity, the concept is similar, with the values of the function approaching a certain limit as the independent variable becomes very large.

Logarithmic functions are inverses of exponential functions and play a crucial role in various areas of mathematics, including the study of slowly varying functions. The function L(t) = log t, where 'log' denotes the natural logarithm, is a classic example of a logarithmic function.

These functions increase very slowly compared to polynomial or exponential functions, which is why they are particularly interesting when studying asymptotic behavior. In the exercise, the slowly varying nature of log t is revealed by the limit \( \lim_{t \to\infty} \frac{\log(tx)}{\log(t)} \) being equal to 1 for all positive values of x. This inherent property, that the ratio of the function value at tx to the value at t approaches 1 as t grows, underscores the modest pace at which logarithmic functions grow.

Power functions, such as f(t) = t^ε, where ε is a non-zero constant, show different characteristics compared to logarithmic functions, especially in the context of the 'slowly varying' property. Power functions involve the variable raised to a certain power and can grow or decay rapidly depending on that power.

In the case of the exercise, it is demonstrated that if ε is not equal to zero, the power function t^ε fails to qualify as slowly varying. This stems from the fact that the limit \( \lim_{t \to\infty} (tx)^{ε} / t^{ε} \) simplifies to x^ε, which notably does not approach 1 unless ε is zero. The power ε effectively dictates the rate at which the function escalates, leading to a clear distinction from the behavior of logarithmic functions.

Asymptotic analysis is a method of describing limiting behavior, and it's particularly useful in mathematics and computer science for analyzing algorithms. When it comes to functions, 'asymptotic' generally refers to the tail behavior of functions as the variable grows large.

In the context of the exercise, asymptotic analysis is applied to compare the definitions of slowly varying functions, like L(t) = log t, against non-slowly varying ones, such as t^ε. The idea is to see how each function behaves as we take t to infinity. The analysis reveals that the logarithmic function approaches a constant ratio of 1 asymptotically, whereas the power function does not, which results in distinctly different asymptotic behaviors. Understanding this concept is crucial for students when they analyze the long-term growth or decay rates of different functions.