LIMIT _{x----> 0}(1/sinx)/(1/sinx)

well given answer is limit does not exist explain please

To understand why the limit of the expression \((1/\sin x)/(1/\sin x)\) as \(x\) approaches 0 does not exist, we need to break down the components of the expression and analyze their behavior near that point.

Analyzing the Expression

The expression can be simplified as follows:

• First, rewrite the limit: \(\lim_{x \to 0} \frac{1/\sin x}{1/\sin x}\).
• This simplifies to \(\lim_{x \to 0} 1\), since any non-zero number divided by itself is 1.

Behavior of \(\sin x\) Near Zero

However, the crux of the issue lies in the behavior of \(\sin x\) as \(x\) approaches 0. The sine function approaches 0 when \(x\) is 0:

• \(\sin(0) = 0\)
• As \(x\) approaches 0 from either side (positive or negative), \(\sin x\) gets very close to 0.

Understanding the Limit's Non-Existence

Now, let’s consider the implications of this behavior:

• When \(x\) is very close to 0, \(\sin x\) becomes very small, leading to the terms \(1/\sin x\) approaching infinity.
• Thus, as \(x\) approaches 0, both the numerator and denominator of our original expression approach infinity.

Indeterminate Form

This situation creates an indeterminate form of \(\frac{\infty}{\infty}\). In calculus, limits that result in indeterminate forms require further analysis, often using techniques like L'Hôpital's Rule or algebraic manipulation.

Conclusion on the Limit

In this case, since both the numerator and denominator approach infinity as \(x\) approaches 0, we cannot definitively say that the limit exists. Instead, we conclude that the limit does not exist because it leads to an undefined situation where we cannot determine a specific value.

To summarize, while the expression simplifies to 1 in a straightforward manner, the underlying behavior of \(\sin x\) as it approaches 0 introduces complexities that result in the limit being undefined. This highlights the importance of examining the behavior of functions near critical points when evaluating limits.