The inequality $-1 < \sin(x) < 1$ does not directly invoke the Squeeze Theorem because it's not a statement about limits in the conventional sense. Instead, it's simply a bound that holds for all real numbers $x$, given that the sine function is always constrained between $-1$ and $1$.

Squeeze Theorem

The Squeeze Theorem states that if:

1. $g(x) \leq f(x) \leq h(x)$ for all $x$ near a point $a$ (except possibly at $a$),
2. $\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L$,

then, $\lim_{x \to a} f(x) = L$.

In your example of $-1 < \sin(x) < 1$, the functions $-1$ and $1$ are constants, and they serve as the bounds for $\sin(x)$, but no limit comparison is being made to determine the behavior at a particular point. Instead, $\sin(x)$ simply oscillates between $-1$ and $1$, never exceeding these bounds.

Why Limit Comparison Doesn’t Work Here:

• For the Squeeze Theorem to work, the upper and lower bounds must converge to the same value (limit) at a specific point. However, in this case:
  • $\lim_{x \to a} -1 = -1$,
  • $\lim_{x \to a} 1 = 1$.

Since these limits are not equal, the Squeeze Theorem is not applicable in this context.

Instead, $-1 < \sin(x) < 1$ is just the natural range of the sine function, a fact that holds for all $x$, and not a limit statement.

Summary

The inequality $-1 < \sin(x) < 1$ doesn't conform to the Squeeze Theorem because the upper and lower bounds $-1$ and $1$ are constant, and their limits do not match. The Squeeze Theorem applies when the bounds converge to the same value at a particular point.

Would you like further clarification or details on this? Here are five related questions to deepen understanding:

1. What is an example of the Squeeze Theorem applied to a limit problem?
2. Can $\sin(x)$ be used in a Squeeze Theorem scenario for specific limits?
3. How does the range of sine and cosine functions compare to other periodic functions?
4. What happens to the sine function as $x \to \infty$?
5. What are common applications of the Squeeze Theorem in calculus?

Tip: The Squeeze Theorem is often used to evaluate limits involving trigonometric functions, especially when the function oscillates or has complex behavior near a certain point.