91Ó°ÊÓ

A piecewise function is a function that is defined by different expressions based on various intervals or conditions. In the problem we're discussing, the function \(f(x, y)\) behaves differently depending on whether \(x^2 + y^2\) equals 1 or not. This means that the function is divided into two separate expressions:

• If \(x^2 + y^2 eq 1\), the function is \(\frac{\sin (x^2+y^2-1)}{x^2+y^2-1}\).
• If \(x^2 + y^2 = 1\), the function becomes some constant \(b\).

To understand piecewise functions, it's important to consider the points where the expressions "switch." In this example, the crucial boundary condition is \(x^2 + y^2 = 1\). Across different conditions, piecewise functions can represent scenarios where behavior abruptly changes, such as in physics or real-world data. For continuity, the switch point's limit must match the defined value, which is why finding the appropriate \(b\) is necessary.

Transforming a problem into polar coordinates can often simplify complex expressions, especially when dealing with circular boundaries, such as \(x^2 + y^2 = 1\). By using polar coordinates:

• We write \(x = r\cos(\theta)\) and \(y = r\sin(\theta)\).
• This leads to \(x^2 + y^2 = r^2\), which neatly simplifies the expression.

For the function given, we are interested in what happens as \(r^2\) approaches 1. The boundary condition \(x^2 + y^2 = 1\) translates to \(r^2 = 1\) in polar terms. Switching to polar coordinates can be especially useful because it reduces the problem to a single variable, \(r\), making it easier to handle, particularly in cases requiring limits of expressions that have circular symmetry.

L’Hôpital’s Rule is an essential tool in calculus used to evaluate limits that result in indeterminate forms, like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). In our exercise, the function inside the limit becomes an indeterminate form when \(r^2 = 1\). To resolve this, we employ L'Hôpital's Rule, which allows us to differentiate the numerator and the denominator:

• For \(\sin(r^2 - 1)\), its derivative with respect to \(r\) is \(2r\cos(r^2 - 1)\).
• For \(r^2 - 1\), its derivative is \(2r\).

Applying L'Hôpital’s Rule simplifies the original limit into a more manageable form:\[\lim_{r^2 \to 1} \frac{2r\cos(r^2 - 1)}{2r}\]This expression simplifies further to \(\lim_{r^2 \to 1} \cos(r^2 - 1)\), allowing us to substitute and find \(b = 1\). Using this rule can make solving seemingly complicated limits much more straightforward, and it’s a powerful technique worth mastering.