Questions: lim as x approaches 0 of sin(1 / x)

Solution Steps

To find the limit of \(\sin(1/x)\) as \(x\) approaches 0, we need to consider the behavior of the function. As \(x\) approaches 0, \(1/x\) becomes very large, causing \(\sin(1/x)\) to oscillate between -1 and 1 without settling at any particular value. Therefore, the limit does not exist.

We need to evaluate the limit \( \lim_{x \rightarrow 0} \sin(1/x) \). As \( x \) approaches 0, the term \( 1/x \) approaches infinity, causing \( \sin(1/x) \) to oscillate between -1 and 1.

Since \( \sin(1/x) \) oscillates indefinitely without approaching a specific value as \( x \) approaches 0, we conclude that the limit does not exist. The function does not settle at any particular value, and instead, it continues to oscillate.

The output indicates that the limit is bounded between -1 and 1, but does not converge to a single value. Therefore, we state that the limit does not exist.

Final Answer