91Ó°ÊÓ

The concept of a limit helps us understand how a function behaves as its input approaches a certain value. In our exercise, we're focusing on the limit of the function \(x^2\) as \(x\) approaches 3. The statement \( \lim_{x \rightarrow 3} x^2 = 9 \) means that as \(x\) gets closer and closer to 3, the value of \(x^2\) gets closer and closer to 9.

In the framework of the epsilon-delta definition of a limit, this is expressed as follows: for every positive \(\varepsilon\) (no matter how small), there exists a corresponding positive \(\delta\) such that whenever \(|x - 3| < \delta\), it follows that \(|x^2 - 9| < \varepsilon\). This implies a close relationship between how much \(x\) is allowed to deviate from 3 (measured by \(\delta\)) and how much \(x^2\) can deviate from 9 (measured by \(\varepsilon\)).

Understanding this concept plays a crucial role in calculus where precise control over the behavior of functions near specific points is often required.

To grasp the epsilon-delta definition of limits from a geometric perspective, imagine the graph of the function \(y = x^2\). The point \((3, 9)\) lies on this curve, and as \(x\) approaches 3, the \(y\)-value approaches 9.

This concept can be visualized geometrically. As \(x\) nears 3, either from the left or right, the height of the curve (\(x^2\)) gets closer to the horizontal line \(y = 9\). We want the difference \(|x^2 - 9|\) to be less than any given \(\varepsilon\).

Using the understanding that \(|x^2 - 9| = |x-3||x+3| < \varepsilon\), we can picture the allowable 'strip' around \(y = 9\) defined by \(\varepsilon\). The largest possible \(\delta\) that confines \(x\) within this strip is \(\delta = \sqrt{9 + \varepsilon} - 3\). This choice allows the smallest changes around the graph's point \((3, 9)\) while maintaining \(|x^2 - 9| < \varepsilon\), ensuring \(y\)-values stay confined to the region close to 9.

To verify the limit definition, namely \( \lim_{x \rightarrow 3} x^2 = 9 \), through calculations, consider the conditions \(|x^2 - 9| < \varepsilon\) and \(|x - 3| < \delta\). The expression \(|x^2 - 9|\) simplifies to \(|x-3||x+3|\), hence controlling \(|x-3|\) is key.

By setting \(\delta = \sqrt{9 + \varepsilon} - 3\), we satisfy the condition \(|x^2 - 9| < \varepsilon\). Plugging \(x = 3 + \delta\) into \(x^2\), we confirm the result: when expanded, \((3+\delta)^2 = 9 + 2\cdot 3 \cdot \delta + \delta^2 = 9 + \varepsilon\). This ensures that the deviation of \(x^2\) from 9 is less than the maximum allowable \(\varepsilon\).

Thus, through this calculation and ensuring that the distance \(x\) deviates from 3 respects the defined \(\delta\), we affirm that the choice of \(\delta\) verifies the limit definition effectively. The conclusion is that our expression reliably makes \( \lim_{x \rightarrow 3} x^2 = 9 \), meeting the epsilon-delta criteria perfectly.