Berechnen Sie mit Hilfe des entsprechenden Grenzwertsatzes aus der gegebenen Bildfunktion \(F(s)\) den Endwert \(f(\infty)\) der zugeh?rigen Originalfunktion \(f(t):\) a) \(F(s)=\frac{1}{s(s+4)}\) b) \(F(s)=\frac{\mathrm{e}^{s}-s-1}{s^{2}}\)

The Laplace Transform is a widely used integral transform in mathematics with applications in engineering and physics. It converts a time-domain function, denoted by \(f(t)\), into a complex frequency-domain function, denoted by \(F(s)\). This is especially useful for solving differential equations and analyzing linear time-invariant systems. The Laplace Transform of a function \(f(t)\) is defined as:
\[ F(s) = \text{L}\big\{f(t)\big\} = \int_{0}^{\text{∞}} e^{-st} f(t) \text{d}t \]
By converting into the \(s\)-domain, solving complex differential equations becomes a matter of algebraic manipulation. When transferring back to the time domain, the inverse Laplace Transform is used.

Limits are a fundamental concept in calculus and analysis used to describe the behavior of a function as its argument approaches a particular value. In the context of the Final Value Theorem, limits enable us to determine the steady-state value of a system.
For instance, to apply the Final Value Theorem, we compute \[ f(∞) = \text{lim}_{s \rightarrow 0} [sF(s)] \]
This requires evaluating the limit of \(sF(s)\) as \(s\) approaches zero. Understanding limit calculations is essential for dealing with indeterminate forms and ensuring the correct application of mathematical rules.

L'Hôpital's Rule is a standard method used to evaluate the limit of a quotient of two functions that result in an indeterminate form, such as 0/0 or ∞/∞. The rule states that if the limits of \(f(s)\) and \(g(s)\) as \(s\) approach a value lead to an indeterminate form, then:
\[ \text{lim}_{s \rightarrow c} \frac{f(s)}{g(s)} = \text{lim}_{s \rightarrow c} \frac{f'(s)}{g'(s)} \]
provided the limit on the right side exists. In the Laplace Transform exercise, L'Hôpital's Rule is used to evaluate a limit that otherwise seems indeterminate. For example:
\[ \text{lim}_{s \rightarrow 0} \frac{e^s - s - 1}{s} \]
Using L'Hôpital's Rule, we differentiate the numerator and the denominator and evaluate the new limit.

Engineering Mathematics involves applying mathematical methods and techniques to solve complex problems in engineering. The use of Laplace Transforms and the Final Value Theorem is a prime example of how mathematics supports engineering solutions. These tools help analyze and design systems, providing a means to:

• Convert differential equations to algebraic equations
• Analyze system stability
• Determine system response over time

Laplace Transforms are particularly useful in control systems, signal processing, and electrical engineering, ensuring engineers can predict and optimize system behavior efficiently.