Understanding Non-Existent Limits

00:00 4 marks

Consider the function \( f(x) = \sin(1/x) \).

Explain why its limit does not exist as \( x \) approaches 0.

Additionally, discuss briefly how limits might fail to exist in cases of unbounded behavior, such as functions exhibiting vertical asymptotes.

Hint Hint

Hint

Recall the formal definition of a limit and consider what happens when a function oscillates or becomes unbounded as it approaches a given point.

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Complete Answer

The limit does not exist because the function does not approach a single value. In the case of \(\sin(1/x)\), the rapid oscillations as \( x \) approaches 0 prevent convergence, while in functions with vertical asymptotes, the unbounded behavior causes the limit to fail to exist.

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