(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

We are given two functions f : [itex]\mathbb{R}^n[/itex] -> [itex]\mathbb{R}[/itex] and g: [itex]\mathbb{R}^n[/itex] -> [itex]\mathbb{R}[/itex]. For every x [itex]\in[/itex] [itex]\mathbb{R}^n[/itex] we define the following:

k(x) = max{f(x), g(x)}

h(x) = min{f(x), g(x)}

The question is:

if lim x-> a k(x) exists and lim x-> a h(x) exists, and the limits are equal, does that imply that lim x->a f(x) exists?

2. Relevant equations

3. The attempt at a solution

Suppose we define f(x) = 1/x and g(x) = x. At x = 0 f(x) is not defined, so g(x) is the minimum and the maximum at x = 0. Therefore k(x) = g(x) and h(x) = g(x). We know that lim x-> 0 h(x) exists and lim x-> 0 k(x) exists, but lim x -> 0 f(x) does not exists.

Is my work here correct or am i wrong in assuming that if f(0) is not defined then g(0) is the maximum and the minimum at x = 0?

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# Homework Help: Problem proving if a limit exists

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