Problem 1 1\. Zeigen Sie mit Hilfe des \((... [FREE SOLUTION] (2024)

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The concept of limits is fundamental to the field of calculus and provides the foundation for understanding continuous functions. A limit describes the value that a function approaches as the input approaches some value.

For example, when we say \( \lim_{{x\to a}} f(x) = L \), we mean that as \(x\) gets closer and closer to \(a\), the function \(f(x)\) gets arbitrarily close to the number \(L\). This doesn't necessarily mean that \(f(a) = L\), but rather that \(f(x)\) will tend towards \(L\) as \(x\) approaches \(a\) from either side (left or right).

In the context of the textbook exercise, calculating the left-hand limit (as \(x\) approaches from the left) and the right-hand limit (as \(x\) approaches from the right) and comparing these with the function value at that point helps determine the continuity of the function at \(x_0\). If all three values are equal, the function is continuous at that point; if not, it has a discontinuity.

A continuous function is one that has no interruptions, jumps, or breaks in its graph. Formally, a function \( f \) is continuous at a point \( x_0 \) if three conditions are met: (1) \( f(x_0) \) is defined, (2) \( \lim_{{x\to x_0}} f(x) \) exists, and (3) \( \lim_{{x\to x_0}} f(x) = f(x_0)\). Essentially, you can draw the graph of a continuous function without lifting your pencil from the paper.

Understanding the continuity of a function is crucial in many areas of mathematics and its applications, including engineering. Continuous functions behave predictably and are often easier to work with than functions that have discontinuities. The textbook examples illustrate this by verifying continuity at given points through the calculation of limits and direct function evaluation.

Piecewise functions consist of multiple sub-functions, each defined on a particular interval. These functions can model complex behavior and are especially useful when the process being described changes its nature depending on the input value.

For a piecewise function to be continuous, each sub-function must be continuous within its interval, and the function must transition smoothly from one sub-function to the next. This means that the left-hand limit and the right-hand limit at the junction point must match the function's value at that point. In the textbook exercise, piecewise functions are analyzed for continuity by evaluating the limits on either side of the specified points and ensuring these limits agree with the function’s value at those points.

Engineering mathematics is an area of applied mathematics that focuses on solving complex engineering problems. It often involves concepts like continuity, limits, and piecewise functions to understand and predict the behavior of physical systems.

Engineers use mathematics to model, analyze, and design everything from bridges to electrical circuits. Functions describing physical phenomena must often be continuous or at least piecewise continuous to match the behavior of the real world. Discontinuities can represent sudden changes in materials, forces, or boundary conditions. The textbook exercises serve as examples of how engineers might confirm that their mathematical models behave as expected across different domains or conditions.