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Differentiable functions are those for which a derivative exists at every point in their domain. In simple terms, the graph of a differentiable function should be smooth and not have sharp corners or discontinuities. For the given functions, both \(f(x) = x^2 \sin(1/x)\) for \(0 < x \leq 1\) and \(g(x) = x^2\) for \(x \in [0,1]\) are differentiable on their respective intervals. A smooth graph indicates that at each point where the function is defined, there's a unique tangent line whose slope corresponds to the derivative of the function. This concept is crucial in calculus as differentiable functions are essential for defining rates of change and ensuring smooth transitions in practical applications such as physics and engineering.
To better grasp differentiable functions, remember these key ideas:

• Smooth and continuous curve
• Exists derivative at every point in its domain
• No sudden jumps or jagged edges

The concept of the limit of a function is foundational in calculus. Limits help us comprehend how functions behave near a particular point, even if they aren't defined at that point. For the given exercise, we examine the limit of \(f(x)\) and \(g(x)\) as \(x\) approaches 0. The function \(f(x)=x^2 \sin(1/x)\) directly illustrates the function's behavior even if it's undefined exactly at \(x = 0\), while \(f(0) = 0\) is defined separately. Substituting the limit, we find \(f(x) = 0 \cdot \sin(\infty) = 0\), interpreted as 0, due to the property that \(\sin(\theta)\) is bounded between \(-1\) and \(1\). Consequently, the limit \(\lim_{x \to 0} f(x) = 0\).

• Limits describe the behavior of functions as they reach specific input values.
• They are especially useful when the function itself isn't directly computable.
• Understanding limits allows for a deeper grasp of the function's continuity and differentiability.
• If both left-hand and right-hand limits exist and equal each other, the limit exists.

Indeterminate forms emerge when evaluating limits yields vague or undefined expressions. A common example is \(0/0\), encountered when computing the limit of \(f(x)/g(x)\) as \(x\) approaches 0 in the given problem. Both \(f(x)\) and \(g(x)\) evaluated at 0 separately lead to zero, thus forming \(0/0\), an indeterminate form. In mathematical analysis, such forms suggest that more information or advanced techniques, like L'Hôpital's Rule, are required to determine the limit's existence or its value. Applying L'Hôpital's Rule involves differentiating the numerator and the denominator separately until a determinable limit is achieved. However, in this exercise, the attempt to resolve \(\lim_{x \to 0} (f(x)/g(x))\) results in \(\sin(1/x)\), which continuously oscillates between \(-1\) and \(1\), further confirming that the limit does not exist due to this persistent fluctuation.

• Indeterminate forms indicate limits requiring special evaluation techniques.
• Common forms include \(0/0\), \(\infty/\infty\), and others.
• They highlight areas where straightforward substitution isn't viable.
• Advanced tools like L'Hôpital's Rule can sometimes resolve these ambiguities.