3. Consider the following definitions: Definition 1 Let f, g: R R. Then, f(x) = O(g(x))...

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3. Consider the following definitions: Definition 1 Let f, g: R R. Then, f(x) = O(g(x)) as { x x xx x a Definition 2 Let f, g: R R. Then, f(x) = o(g(x)) as { x x lim sup if and only if lim supr- lim sup-a lim x x X if and only if {lim- x a - limxa Let n0. Show that 0(x) +0 (x) = 0 (x) as x 0 0(x)+0(x) = 0 (x") as x 0 |f(x) | g(x) |f(x) = 0 g(x) (6) For example, to prove (6), let f(x) = 0 (x) and g(x) = 0 (x^) as x 0, and show that f(x) + g(x) = 0 (x) as x 0, i.e., that |f(x) + g(x) lim sup < xn 0+x 3. Consider the following definitions: Definition 1 Let f, g: R R. Then, f(x) = O(g(x)) as { x x xx x a Definition 2 Let f, g: R R. Then, f(x) = o(g(x)) as { x x lim sup if and only if lim supr- lim sup-a lim x x X if and only if {lim- x a - limxa Let n0. Show that 0(x) +0 (x) = 0 (x) as x 0 0(x)+0(x) = 0 (x") as x 0 |f(x) | g(x) |f(x) = 0 g(x) (6) For example, to prove (6), let f(x) = 0 (x) and g(x) = 0 (x^) as x 0, and show that f(x) + g(x) = 0 (x) as x 0, i.e., that |f(x) + g(x) lim sup < xn 0+x