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Limit analysis is critical when comparing the behavior of two functions as they approach infinity. We use limits to determine if two functions are asymptotic. Remember, two functions \( f \) and \( g \) are asymptotic if: \[ \lim_{{x \to \infty}} \frac{{f(x)}}{{g(x)}} = 1 \] This means that as \( x \) grows larger, the ratio of \( f(x) \) to \( g(x) \) gets closer and closer to 1. If this limit is anything other than 1, the functions are not asymptotic.

Limits help us simplify complex expressions. For instance, in the first example \[ f(x) = \log(x^{2} + 1) \] and \[ g(x) = \log x \], evaluating the limit \( \lim_{{x \to \infty}} \frac{{\log(x^{2} + 1)}}{{\log x}} \) requires approximations and simplifications to make calculation manageable.

Logarithmic functions are essential in calculus and limit analysis. They have the form \( \log_a(x) \), which means the power to which a base, \( a \), must be raised to obtain \( x \). A special property of logarithms is that they turn multiplication into addition: \[ \log_a(xy) = \log_a(x) + \log_a(y) \] and powers into multiplications: \[ \log_a(x^n) = n \log_a(x) \]

In our exercise, one function was \( \log(x^{2} + 1) \). For large \( x \), \( x^{2} + 1 \) is nearly the same as \( x^2 \), allowing us to use the property of logarithms to simplify: \( \log(x^{2} + 1) \approx \log(x^{2}) = 2 \log(x) \). This simplification plays a key role in evaluating the limit.

Exponential functions have the form \( f(x) = a^x \), where \( a \) is a constant. These functions grow or decay at rates proportional to their current value. For example, \( 2^x \) grows very quickly as \( x \) increases, much faster than any polynomial or logarithmic function.

In parts (b), (c), and (d) of the exercise, we dealt with various exponential functions. For instance: \( 2^{x+3} \) and \( 2^{x+7} \). By simplifying their ratio, we use properties such as \[ \frac{{2^{x+3}}}{{2^{x+7}}} = 2^{(x+3)-(x+7)} = 2^{-4} \] Understanding these properties allows us to easily evaluate the ratios and their limits, making these calculations straightforward if approached correctly.

Asymptotic behavior describes how functions behave as \( x \) approaches infinity. This is crucial for understanding if two functions are asymptotic. If two functions \( f(x) \) and \( g(x) \) are asymptotic, it means they grow or decay similarly at infinity, which can be formalized by the limit of their ratio approaching 1.

For example, in the last part (d) of the exercise with \( f(x) = 2^{x^{2}+x+1} \) and \( g(x) = 2^{x^{2}+2x} \), we want to see how these functions compare at very large \( x \). By simplifying the expression \( \frac{{2^{x^{2}+x+1}}}{{2^{x^{2}+2x}}} = 2^{(x^{2}+x+1)-(x^{2}+2x)} = 2^{1-x} \), we observe that as \( x \to \infty \), \( 2^{1-x} \to 0 \). This demonstrates they are not asymptotic as the limit approaches 0, not 1.

Understanding asymptotic behavior helps predict long-term trends and make sense of complex functions.