Question

Find differentiable functions \(f\) and \(g\) such that \(\lim _{x \rightarrow \infty} f(x)=\lim _{x \rightarrow \infty} g(x)=\infty \quad\) and \(\lim _{x \rightarrow \infty}[f(x)-g(x)]=25\). Explain how you obtained your answers. (Note: There are many correct answers.)

Short answer

The functions are \(f(x) = x + 25\) and \(g(x) = x\).

Step-by-step solution

Identify appropriate functions

To solve this problem, let's consider simple linear functions. Based on the given conditions, one could take \(f(x)\) as \(x + c\) and \(g(x)\) as \(x\), where \(c\) is a constant.

Find the constant

Given that \(\lim _{x \rightarrow \infty}[f(x)-g(x)]=25\), substituting our chosen functions we get: \(\lim _{x \rightarrow \infty}[(x + c) - x] = 25\). From this, we can see that \(c=25\).

Check the conditions

Now that we've found \(c\), let's plug it back into our functions: \(f(x) = x+25\), \(g(x) = x\). It is clear that both functions are differentiable everywhere. Also, as \(x\) becomes large, both \(f(x)\) and \(g(x)\) tend to infinity. Therefore, they meet all the set conditions.