Question

In Exercises \(59 - 62 ,\) find the limit graphically. Use the Sandwich Theorem to confirm your answer. $$\lim _ { x \rightarrow 0 } x \sin x$$

Short answer

The limit of \(x \sin x\) as \(x\) approaches \(0\) is \(0\).

Step-by-step solution

Graphical Representation

Draw a graph of the function \(x \sin x\) as \(x\) approaches \(0\). You will find that the limit as \(x\) approaches \(0\) is \(0\).

Understanding the Limit

From graph, it can be observed that as \(x\) approaches \(0\), the value of \(x \sin x\) also approaches \(0\). So, the limit of \(x \sin x\) as \(x\) approaches \(0\) is \(0\).

Application of the Sandwich Theorem

The Sandwich Theorem states that if for all \(x\), \(f(x) \leq g(x) \leq h(x)\) and the limit as \(x\) approaches \(a\) of \(f(x)\) and \(h(x)\) are equal, then the limit of \(g(x)\) as \(x\) approaches \(a\) is also equal to this common limit. We can say that \(x \sin x\) is sandwiched between \(-x\) and \(x\) because \(-x \leq x \sin x \leq x\). As \(x\) approaches \(0\), the limit of \(-x\) and \(x\) is \(0\), which is the same as the limit of \(x \sin x\), hence confirming the answer through Sandwich Theorem.