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One of the powerful tools in calculus is the Squeeze Theorem, also known as the Sandwich Theorem. It's particularly helpful when you want to find the limit of a function that is difficult to evaluate directly. Imagine you're "squeezing" a function between two others that are easier to analyze.
Here's how it works:- If you have three functions, say \( g(x) \), \( f(x) \), and \( h(x) \), and they satisfy \( g(x) \leq f(x) \leq h(x) \) over some interval.- If \( \lim\limits_{x \to a} g(x) = L \) and \( \lim\limits_{x \to a} h(x) = L \), where both limits are equal.- Then, thanks to the Squeeze Theorem, \( \lim\limits_{x \to a} f(x) = L \) as well.In our exercise, we squeezed the function within the bounds defined by a quadratic expression. By demonstrating that \(-h^2 \leq f(h) \leq h^2\), we were able to prove the limit of \( \frac{f(h)}{h} \) as \( h \to 0 \) was 0. This technique is effective when functions approach from above and below towards the same limit.

In calculus, limits help us understand the behavior of a function as it approaches a certain point or value. They are crucial when exploring continuity and differentiability.
A simple way to picture a limit is to imagine drawing closer and closer to a specific point, observing the function's output. If the function tends toward a particular value as you approach this point, that's the limit of the function at that point.There are critical properties and rules helpful in computing limits:- Limit of sums: \( \lim{(a+b)} = \lim{a} + \lim{b} \)- Limit of products: \( \lim{(a \cdot b)} = \lim{a} \cdot \lim{b} \)- Limit of quotients: \( \lim{\left(\frac{a}{b}\right)} = \frac{\lim{a}}{\lim{b}} \) (provided \( \lim{b} eq 0 \))In part (b) of our exercise, the limit is used to establish the differentiability of \( f(x) \) at \( x = 0 \). We looked at \( \lim\limits_{h \to 0} \frac{f(h) - f(0)}{h} \), determining it simplifies to a limit that evaluates to zero, confirming differentiability.

Derivatives provide a way to measure how a function changes as its input changes. Put simply, a derivative tells us the rate of change or the slope of the function at a given point. The process of finding a derivative is called differentiation.
The formal definition of a derivative at a point \( x = a \) is the limit:\[ f'(a) = \lim\limits_{h \to 0} \frac{f(a + h) - f(a)}{h} \]It represents the instantaneous rate of change of the function as \( h \), a small increment in \( x \), approaches zero.In differentiability analysis, particularly in this exercise, we've applied this definition:- For \( f \) to be differentiable at a point \( x = 0 \), the involved limit must exist.- As seen, both expressions in parts (a) and (b) were handled similarly, leading to the conclusion that \( f'(0) = 0 \).This shows how derivatives help us explore and conclude differentiability, capturing the essential behavior of functions at points of interest.

Piecewise functions are those made up of multiple sub-functions, each defined over different intervals of the domain. They can be handy for modeling real-world scenarios where a relationship changes based on the input value.
Each "piece" of such a function is usually depicted using a specific formula for its range segment. A classic marker of piecewise functions is conditions (usually inequalities) that specify which formula applies to which range.
For example, in this exercise, the function \( f(x) \) is defined as:- \( x^2 \sin\left(\frac{1}{x}\right) \) for \( x eq 0 \)- \( 0 \) when \( x = 0 \)Understanding how piecewise functions transition between different sub-functions is key when computing limits and derivatives. For differentiability, especially at points where the defining formula changes.Here, we looked at the behavior around \( x = 0 \) to check if the transition was smooth enough for differentiability, relying heavily on the behavior of each piece near the transition point.