Question

Writing to Learn Explain why there is no value \(L\) for which \(\lim _{x \rightarrow \infty} \sin x=L\)

Short answer

There is no value \(L\) for which \(\lim _{x \rightarrow \infty} \sin x=L\), because the sine function does not approach a specific number as \(x\) increases indefinitely, but continues to oscillate between -1 and 1.

Step-by-step solution

Recognize the Sine Function

The function \(\sin x\) is moderately difficult to analyze, because it does not tend towards a limit as \(x\) approaches infinity. Here, a deep understanding of the sine function is required, which is a wave pattern that oscillates between -1 and 1.

Understand the Limit Concept

A limit as \(x\) approaches infinity of a function equals \(L\) only if the values of the function get arbitrarily close to \(L\) as \(x\) becomes larger and larger (i.e. approaches infinity). Essentially, this means that the function begins to flatline as \(x\) becomes extremely large, and it will continually get closer and closer to \(L\).

Analyze the Combination of Sine function and Limit Property

The sine function does not flatline or get arbitrarily close to any single number as \(x\) approaches infinity. Instead, it continues to oscillate indefinitely. So there is no single limit \(L\) as \(x\) approaches infinity.