Squeeze Theorem

The squeeze theorem, also known as the pinching theorem or the sandwich theorem, is a fundamental theorem in calculus and real analysis. It provides a method to evaluate the limit of a function by comparing it to two other functions that "squeeze" the original function.

Statement of the Theorem

Suppose that we have three functions $f$, $g$, and $h$ such that $f(x) \leq g(x) \leq h(x)$ for all values of $x$ in some interval $I$, except possibly at the point $x=a$. If $\lim_{x \to a} f(x) = \lim_{x \to a} h(x) = L$, then $\lim_{x \to a} g(x) = L$.

In other words, if the functions $f$ and $h$ approach the limit $L$ as $x$ approaches $a$, and the function $g$ is always squeezed between them, then $g$ must also approach the same limit $L$ as $x$ approaches $a$.

More formally, we can write the squeeze theorem as follows:

$\lim_{x \to a} f(x) = \lim_{x \to a} h(x) = L \;\;\;\; \text{and} \;\;\;\; f(x) \leq g(x) \leq h(x) \;\;\;\; \text{for all} \;\;\;\; x \in I \setminus \{a\}$

$\Rightarrow \lim_{x \to a} g(x) = L$

Example

Let's consider the function $f(x) = x^2$, $g(x) = x\sin\left(\frac{1}{x}\right)$, and $h(x) = x^2$ again. We will use the squeeze theorem to evaluate $\lim_{x \to 0} g(x)$.

First, we need to show that $f(x) \leq g(x) \leq h(x)$ for all $x \neq 0$. It is clear that $f(x) = x^2 \leq x\sin\left(\frac{1}{x}\right) \leq x^2 = h(x)$ for all $x \neq 0$.

Since $\lim_{x \to 0} x^2 = 0$, we have $\lim_{x \to 0} f(x) = 0$. Similarly, $\lim_{x \to 0} h(x) = 0$.

Therefore, by the squeeze theorem, we have:

$\lim_{x \to 0} g(x) = 0$

Importance of the Squeeze Theorem

The squeeze theorem is an important result in calculus and real analysis because it provides a way to evaluate the limit of a function that is not easily calculable directly. It also allows us to prove the limit of a function exists when it is not obvious.

For example, one can use the squeeze theorem to show that the limit of $\sin(x)/x$ as $x$ approaches 0 exists and is equal to 1. This is an important result in calculus, as it is used to derive several other important limits, such as $\lim_{x \to 0}\frac{1-\cos(x)}{x}=0$.

Conclusion

The squeeze theorem is a powerful tool in calculus and real analysis for evaluating the limit of a function when it is not easily calculable. It provides a systematic approach to proving the existence of limits and is used to derive several important limits.

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