Question

Find two functions \(f\) and \(g\) such that \(\lim _{x \rightarrow 0} f(x)\) and \(\lim _{x \rightarrow 0} g(x)\) do not exist, but \(\lim _{x \rightarrow 0}[f(x)+g(x)]\) does exist.

Short answer

Two suitable functions which fulfil the criteria are \(f(x) = \frac{1}{x}\) and \(g(x) = -\frac{1}{x}\). Each of these functions, on their own, do not have a limit as 'x' approaches 0, but when they are combined, the sum \(f(x)+g(x)\) equals to 0 for all real 'x', including 'x' = 0, hence the limit exists.

Step-by-step solution

Title

Choose a function 'f' such that its limit does not exist as 'x' approaches 0. A potential function could be \(f(x) = \frac{1}{x}\). The limit does not exist because as 'x' approaches 0, the value of \(f(x)\) tends to infinity.

Title

Choose another function 'g' such that its limit does not exist as 'x' approaches 0, and whose combined value with 'f' for any 'x' will give a number that approaches a constant value as 'x' approaches 0. A potential function could be \(g(x) = -\frac{1}{x}\).Similar to \(f(x)\), the limit of \(g(x)\) as 'x' approaches zero does not exist because the function tends towards negative infinity.

Title

Combing the two chosen functions 'f' and 'g', the combined function l(x) can be written as \(l(x) = f(x) + g(x) = \frac{1}{x} - \frac{1}{x} = 0\). As 'x' approaches 0, the combined function \(l(x) = 0\), thus \(\lim _{x \rightarrow 0} l(x) = 0\) which exists.