Question

Use a graphing utility to graph the given function and the equations \(y=|x|\) and \(y=-|x|\) in the same viewing window. Using the graphs to visually observe the Squeeze Theorem, find \(\lim _{x \rightarrow 0} f(x)\). $$ f(x)=|x| \sin x $$

Short answer

The limit of the function \(f(x) = |x| \sin x\) as \(x\) approaches 0 is 0.

Step-by-step solution

Graph the function and the limiting equations

Using a graphing utility, graph the function \(f(x) = |x| \sin x\), and the equations \(y = |x|\) and \(y = -|x|\). Ensure all three are in the same viewing window.

Observe the Squeeze Theorem visually

From graph, observe that regardless of the value of \(x\), the function \(f(x) = |x| \sin(x)\) is always squeezed between \(y = |x|\) and \(y = -|x|\). This means, for any value of \(x\), \( -|x| \leq f(x) \leq |x|\). This visual inspection confirms the Squeeze Theorem.

Compute the limit

Observe the graph as \(x\) approaches 0. Note that both \(y = |x|\) and \(y = -|x|\) also approach 0 as \(x \rightarrow 0\). Because \(f(x)\) is squeezed between these two, the limit of \(f(x)\) as \(x\) approaches 0 must also be 0. We can write this formally as \(\lim _{x \rightarrow 0} f(x) = 0\)