91Ó°ÊÓ

A piecewise function is a type of function defined by different expressions based on intervals of the input value. In other words, the function has a "piece" dedicated to certain sections of the number line. In the given exercise, the function is defined differently for values of \(x\) that are less than 0 and for values that are greater than or equal to 0. For \(x < 0\), the function \(f(x)\) is defined as \(e^x\), an exponential decay function. For \(x \geq 0\), the function is \(f(x) = x^2\), a simple quadratic function that forms a parabola.
These two "pieces" are connected at the point \(a = 0\). Understanding piecewise functions is crucial because each segment can represent very different behaviors and contribute to discontinuities.

Limits are a fundamental concept in calculus, helping us analyze how functions behave as they approach certain points or infinity. In our exercise, the limit is used to determine the behavior of the piecewise function as \(x\) approaches \(a = 0\).
Calculating a limit involves assessing the value that a function gets closer to as the input approaches a particular value. There are two key limits to consider in this scenario: the left-hand limit (for approaching from the left) and the right-hand limit (for approaching from the right). Getting comfortable with limits allows us to analyze smooth transitions or jumps in a function's graph.

Discontinuity occurs when there is an abrupt change in the value of a function, creating a "jump" or "break" at a particular point. For the given piecewise function, we check for discontinuity at \(x = 0\) by comparing limits. A function is continuous at a point if the left-hand limit, the right-hand limit, and the function's value at that point are all equal.
In our example, the left-hand limit and right-hand limit at \(x = 0\) are 1 and 0, respectively, leading to a discontinuity. The graph will show a clear jump at this point. Recognizing discontinuities can help in various applications, such as understanding sudden changes in physical systems.

The left-hand limit of a function as \(x\) approaches a specific point, \(a\), is the value that \(f(x)\) approaches as \(x\) gets close to \(a\) from the left side (values \(x < a\)). For the piecewise function in the exercise, as \(x\) approaches 0 from the left (\(x < 0\)), the function \(f(x) = e^x\) dictates the behavior. The left-hand limit is calculated as \(\lim_{x \to 0^-} e^x = e^0 = 1\).
This limit assists in determining the nature of the function's continuity at \(a = 0\). If the left and right-hand limits are not equal, it indicates a discontinuity.

The right-hand limit involves approaching a point \(a\) from the right side (values \(x > a\)). It is crucial when evaluating piecewise functions and their potential discontinuities. For our function, as \(x\) approaches 0 from the right, the expression \(f(x) = x^2\) becomes significant. Therefore, the right-hand limit is calculated as \(\lim_{x \to 0^+} x^2 = 0\).
Comparing the right-hand limit with the left-hand limit and the function's value at \(a = 0\) uncovers any discrepancies that denote discontinuity. In this case, the right-hand limit and the function's value at \(0\) are equivalent, but this does not align with the left-hand limit, confirming a jump at \(x = 0\).