Calculus Limits and Notations

Study Notes

Limit notation: 𝑥 approaches 𝑎 from the left is denoted as 𝑥 → 𝑎−, and from the right as 𝑥 → 𝑎+.
Distinction: −𝑎 and +𝑎 represent numerical values, while 𝑎− and 𝑎+ appear in limit contexts only.

Left limit: denoted as lim 𝑓(𝑥) as 𝑥 → 𝑎− and right limit as lim 𝑓(𝑥) as 𝑥 → 𝑎+.
If left limit ≠ right limit, then two-sided limit does not exist (DNE).
Example cases:
- If lim− 𝑓(𝑥) = 1 and lim+ 𝑓(𝑥) = 3, then lim 𝑓(𝑥) = DNE.
- If lim− 𝑓(𝑥) = lim+ 𝑓(𝑥) = 2, then lim 𝑓(𝑥) exists and equals 2.

Graphical Method: Visual representation to observe function behavior near point.
Numerical Method: Evaluate function values at points close to 𝑎.
Algebraic Method: Use algebra to manipulate and simplify function expressions to find limits.

For lim 𝑥→0 (sin 𝑥)/𝑥, both one-sided limits equal 1, leading to the conclusion that lim 𝑥→0 (sin 𝑥)/𝑥 = 1.
Limit of 𝑓(𝑥) = (𝑥−1)/(𝑥−1) as 𝑥 approaches 1 results in undefined at 𝑥 = 1 but lim− and lim+ both equal 2; thus lim 𝑥→1 𝑓(𝑥) = 2.

Example: lim 𝑥→0 (1/𝑥) pushes limits to negative and positive infinity: lim− = −∞ and lim+ = ∞. Thus, lim 𝑥→0 (1/𝑥) = DNE.
Cases of infinite limits illustrate behavior approaching vertical asymptotes, leading to consistent increases or decreases without bounds.

Vertical asymptotes occur where limits trend to ±∞, indicating areas where the function approaches undefined behavior.
For example, 𝑥 = 0 is a vertical asymptote for 𝑓(𝑥) = 1/𝑥, as the function exhibits an infinite discontinuity at that point.

Economies of developed nations rely heavily on software, with increasing systems being software-controlled.
Software engineering focuses on theories, methods, and tools for professional software development.
Significant software expenditure is evident in the Gross National Product (GNP) of developed countries.

Generic Products:
- Stand-alone systems marketed to any customer.
- Examples include PC software like graphics programs, project management tools, and CAD software.
Customized Products:
- Created on commission to meet specific customer needs.
- Examples are embedded control systems, air traffic control software, and traffic monitoring systems.

Generic Products:
- Specification control lies with the software developer, who decides on changes.
Customized Products:
- Specification is owned by the customer, who dictates required changes.

Maintainability:
- Software should evolve to meet changing customer needs, crucial in dynamic business environments.
Dependability and Security:
- Encompasses reliability, safety, and protection against malicious usage.
Efficiency:
- Software must optimize resource usage such as memory and processing power, impacting responsiveness and processing time.
Acceptability:
- Software must be user-friendly, understandable, and compatible with existing systems for its target audience.

Covers all software production aspects, from system specification to post-deployment maintenance.
Involves problem-solving through appropriate theories and methods, considering organizational and financial constraints.
Encompasses technical development and project management, including the evolution of supporting tools and methodologies.

Software Specification:
- Definition of software requirements and operational constraints by customers and engineers.
Software Development:
- Involves designing and programming the software.
Software Validation:
- Checking the software to ensure it meets customer requirements.
Software Evolution:
- Modifications made to the software in response to changing customer and market demands.

Stand-alone Applications:
- Run locally on computers, containing all necessary functionalities without the need for network connection.
Interactive Transaction-based Applications:
- Executed on remote servers and accessed from user PCs or terminals; includes web applications, like e-commerce.
Embedded Control Systems:
- Software systems that manage hardware devices, likely the most numerically predominant application type.