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The Squeeze Theorem, also known as the Sandwich Theorem or the Pinching Theorem, is a fundamental concept in calculus that asserts if a function is squeezed or sandwiched between two other functions at a certain point and if these two functions converge to the same limit, then the 'squeezed' function must also converge to that same limit.

For example, given three functions, if we have \( f(x) \leq g(x) \leq h(x) \) and the limits of \( f(x) \) and \( h(x) \) are equal as \( x \) approaches \( a \), that is, if \( \lim_{{x \rightarrow a}} f(x) = \lim_{{x \rightarrow a}} h(x) = L \), then \( \lim_{{x \rightarrow a}} g(x) = L \) as well. This powerful tool is particularly useful for evaluating limits where direct substitution or algebraic manipulation isn't possible.

The concept of limits at infinity pertains to the behavior of functions as the variable \( x \) grows without bounds, either positively or negatively. Mathematically, we express this as \( x \) approaches infinity (\( \infty \) or \( -\infty \)).

The main focus here is to understand what value the function converges to, if any, as the input grows larger and larger. When \( \lim_{{x \rightarrow \infty}} f(x) = L \) or \( \lim_{{x \rightarrow -\infty}} f(x) = L \) for some real number \( L \), it means that the values of \( f(x) \) get arbitrarily close to \( L \) as \( x \) becomes very large or very small in the negative direction. This concept is particularly relevant when discussing horizontal asymptotes of functions.

Trigonometric functions, such as sine \( (\sin) \), cosine \( (\cos) \), and tangent \( (\tan) \) are fundamental in calculus, especially when dealing with periodic behaviors and harmonics. Each trigonometric function has specific properties and values that range over certain intervals.

For instance, the cosine function ranges between -1 and 1 for all real numbers, meaning \( -1 \leq \cos(x) \leq 1 \). This boundedness property is enormously beneficial when applying the Squeeze Theorem because it allows functions involving trigonometric components to be estimated and their limits easily found, as demonstrated in the original exercise where the cosine function set the bounds for squeezing.

Inequalities play a significant role in calculus by providing a framework for comparing the relative sizes of functions and determining their behavior. When solving inequalities, it's crucial to remember that the standard rules for inequalities apply, just as they do in algebra.

However, when encountering limits in the context of inequalities, it is important to maintain the direction of the inequality while evaluating the limits separately. If \( f(x) \leq g(x) \) for all \( x \) in some interval, and we take the limit as \( x \) approaches a point within that interval, the result must preserve the inequality, like \( \lim_{{x \rightarrow a}} f(x) \leq \lim_{{x \rightarrow a}} g(x) \) assuming the limits exist. This allows calculus students to solve problems involving limits of functions where direct computation is not possible, as in the case of the textbook exercise involving trigonometric functions and limits at infinity.