Q: Problem 2: Determine the limit of ##f(x) = (x^2 + 1)/x^3## as ##x \to \infty##.

The limit is 0 because the degree of the denominator is greater than the degree of the numerator.

Q: Problem 3: Investigate the convergence of ##\sum_{n=1}^{\infty} \frac{1}{n^2}##.

The series converges to ##\frac{\pi^2}{6}##, a well-known result from calculus.

Q: Problem 4: Evaluate ##\lim_{x \to 0} \frac{\sin(x)}{x}##.

The limit is 1, a fundamental result in calculus often proven using L’Hôpital’s rule or geometric arguments.

Q: Problem 5: Analyze the behavior of ##f(n) = n!## for large ##n##.

Stirling’s approximation provides an accurate estimate: ##n! \approx \sqrt{2\pi n} \left(\frac{n}{e}\right)^n##.

Question 1

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Exploring the behavior of functions is a fundamental aspect of mathematics and science. Often, we analyze functions using small or common numbers to identify patterns. However, this approach can be misleading when dealing with very large numbers. This article delves into functions that exhibit significantly different behavior at large scales compared to their behavior at smaller scales, highlighting potential pitfalls and warning signs.

The Griewank Function and Scale

The Griewank function, a common test case for optimization algorithms, provides a striking example of scale-dependent behavior. Defined as ### f(\mathbf x) = \frac1{4000}\sum_{i=1}^n x_i^2 &#8211; \prod_{i=1}^n \cos\left(\frac{x_i}{\sqrt i}ight) + 1 ###, this function&#8217;s behavior is drastically different at small and large scales. Understanding the Griewank function helps illustrate the challenges in predicting function behavior across different scales.

At a small scale, the cosine product term dominates the Griewank function, leading to oscillatory behavior. However, as the values of ##x_i## increase, the quadratic term ##\frac1{4000}\sum_{i=1}^n x_i^2## becomes dominant. This shift causes the function to behave more like a quadratic, masking the oscillatory nature seen at smaller scales. This illustrates how initial observations can be misleading when extrapolating to very large numbers, particularly in optimization contexts.

Rational Functions and Asymptotic Behavior

Rational functions, expressed as ##f(x) = \frac{n(x)}{d(x)} = q(x) + \frac{r(x)}{d(x)}## where deg(r) < deg(d), often behave differently near the zeros of their denominators compared to their behavior at large ##x## values. Near the zeros of ##d(x)##, the term ##\frac{r(x)}{d(x)}## dominates, causing the function to exhibit singular or asymptotic behavior. Understanding the behavior of rational functions requires careful consideration of both local and global properties.

For large positive or negative values of ##x##, the quotient ##q(x)## dominates the behavior of the rational function. The term ##\frac{r(x)}{d(x)}## diminishes as ##x## increases, and ##f(x)## begins to resemble ##q(x)##. This asymptotic behavior is crucial in many applications, such as approximating functions or analyzing limits. Therefore, analyzing rational functions requires distinguishing between local singularities and overall trends.

Chebyshev Bias and Large Number Examination

The Chebyshev bias, a subtle phenomenon in number theory, can only be properly understood by examining extremely large numbers. This bias refers to the observed tendency for prime numbers to have a preference for certain residue classes modulo a given integer. Detecting and understanding this bias requires analyzing prime distributions over vast ranges, far beyond what is typically considered in elementary number theory.

Specifically, the Chebyshev bias illustrates that initial patterns observed in prime number distributions can be misleading. For example, one might initially observe an equal distribution of primes across residue classes. However, as numbers grow larger, subtle biases emerge, revealing underlying structures that are not apparent at smaller scales. Therefore, the Chebyshev bias underscores the importance of considering very large numbers when studying number-theoretic phenomena.

Asymptotic Analysis and Intuition

The values of a function ##f(x)## for relatively small values of its argument ##x## are often poor predictors of the asymptotic behavior of ##f(x)## for large ##x##. This holds true even for analytic functions, which are uniquely determined by their values on any small interval. The behavior of functions at infinity can defy intuition gained from small-scale observations, necessitating rigorous asymptotic analysis.

Asymptotic analysis requires cultivating an expansive attitude, thinking big when imagining a variable approaching infinity. For instance, while ##\log n \prec n^{0.0001}## might seem incorrect for small numbers, it holds true for sufficiently large ##n##. This illustrates the importance of considering extremely large numbers to accurately assess the long-term behavior of functions. Therefore, intuition based on small values can be misleading, emphasizing the need for careful analysis.

Warning Signs and Considerations

When analyzing function behavior, several warning signs indicate that large numbers might cause unexpected behavior. Functions involving oscillations, such as trigonometric functions or series with alternating signs, are particularly prone to scale-dependent behavior. Additionally, functions with singularities or discontinuities can exhibit dramatically different behavior near these points compared to their behavior at large scales.

Furthermore, functions defined by recursive relations or involving number-theoretic concepts often require careful examination at large scales. Prime numbers, in particular, can introduce complexities that defy simple extrapolation from small-scale observations. Therefore, a combination of analytical techniques and computational exploration is often necessary to fully understand function behavior across different scales, especially when dealing with very large numbers.

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