A) Show that if f(x) and g(x) are functions such that f(x)...

a) Show that if $f(x)$ and $g(x)$ are functions such that $f(x)$ is $o(g(x))$ and $c$ is a constant, then $c f(x)$ is $o(g(x))$ where $(c f)(x)=c f(x)$ b) Show that if $f_{1}(x), f_{2}(x),$ and $g(x)$ are functions such that $f_{1}(x)$ is $o(g(x))$ and $f_{2}(x)$ is $o(g(x)),$ then $\left(f_{1}+\right.$ $f_{2} )(x)$ is $o(g(x)),$ where $\left(f_{1}+f_{2}\right)(x)=f_{1}(x)+f_{2}(x)$

a) Show that if $f(x)$ and $g(x)$ are functions such that $f(x)$ is $o(g(x))$ and $c$ is a constant, then $c f(x)$ is $o(g(x))$ where $(c f)(x)=c f(x)$
b) Show that if $f_{1}(x), f_{2}(x),$ and $g(x)$ are functions such that $f_{1}(x)$ is $o(g(x))$ and $f_{2}(x)$ is $o(g(x)),$ then $\left(f_{1}+\right.$ $f_{2} )(x)$ is $o(g(x)),$ where $\left(f_{1}+f_{2}\right)(x)=f_{1}(x)+f_{2}(x)$

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Key Concepts

Little o Notation

Little o notation, denoted as f(x) = o(g(x)), is used to describe the asymptotic behavior of functions. It means that the function f(x) becomes negligible compared to g(x) as x approaches a limit (often infinity), i.e., the limit of f(x)/g(x) is 0. This is a precise way of stating that f(x) grows much slower than g(x) or essentially vanishes in comparison.

Multiplicative Constant Property

This property states that if f(x) is o(g(x)), then multiplying f(x) by any constant c does not change the asymptotic negligibility; that is, c·f(x) is also o(g(x)). The reasoning follows from the fact that scaling by a constant does not affect the limit f(x)/g(x) tending to 0, except possibly when the constant is 0 (in which case the result is trivially 0).

Additive Property of Little o Notation

The additive property asserts that if two functions f1(x) and f2(x) are both o(g(x)), then their sum f1(x) + f2(x) is also o(g(x)). This is because, by linearity of limits, the sum of two functions that are each negligible relative to g(x) will also be negligible relative to g(x). This encapsulates how little o notation is stable under addition.

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