Study Notes on Limits and Functions

Introduction to Limits

• Linear Function, Polynomial Function, and Their Properties
  • Defines basic functions that add values together.
  • Example functions include polynomial forms and constant functions.
• Limit Notation
  • Definition: The limit of a function $f(x)$ as $x$ approaches $c$ is denoted as $\lim_{x \to c} f(x) = l$ where $l$ is a finite value.
• Identity Function: This implies plugging in the value of $c$ is direct and valid for basic functions.

Properties of Limits

• Limit of a Constant Multiple

  • Property: $\lim<em>{x \to c} [k \cdot f(x)] = k \cdot \lim</em>{x \to c} f(x)$ where $k$ is a constant.
  • Example: For $\lim<em>{x \to c} [2x]$, we take out the 2: $2 \cdot \lim</em>{x \to c} x.$

• Limit of Sum/Difference

  • Property: $\lim<em>{x \to c} [f(x) + g(x)] = \lim</em>{x \to c} f(x) + \lim_{x \to c} g(x)$
  • This holds true for subtraction as well.

• Limit of a Product

  • Property: $\lim<em>{x \to c} [f(x) \cdot g(x)] = \lim</em>{x \to c} f(x) \cdot \lim_{x \to c} g(x)$
  • This property allows us to separate products of functions into limits of each function.

• Limit of Quotient

  • Property: $\lim<em>{x \to c} \frac{f(x)}{g(x)} = \frac{\lim</em>{x \to c} f(x)}{\lim_{x \to c} g(x)}$ provided that $g(c) \neq 0$ to avoid division by zero issues.

Special Cases in Limit Evaluation

• Handling Division by Zero
  • Division by zero creates exceptions in limit existences: undefined or potential limit exists.
• Indeterminate Forms (e.g., $\frac{0}{0}$) require special consideration or alternative methods to evaluate.
• Plugging in Values: Basic functions like linear, constant, and polynomial functions can have the value of $c$ plugged in directly.

Examples of Limits

• Evaluating Limits of Polynomial Functions

  • Example: Evaluate $\lim_{x \to 2} (x^3)$ by simply plugging in 2: $(2^3 = 8)$.

• Combining Functions

  • Using properties from above, evaluate:
  • If $f(x) = -x^2 + 1$ and evaluate $\lim_{x \to 1} f(x)$
  • Continue with each term independently and plug in as stated in properties.

• Composite Functions

  • Definition: A composite function is defined as $f(g(x))$
  • To evaluate $\lim_{x \to 4} f(g(x))$:
  • First, evaluate $g(x)$, find the limit, then substitute into $f$ for further evaluation.

Numerical and Graphical Approaches

• Numerical Evaluation
  • When using calculators or numerical methods, ensure proper order of operations to avoid incorrect results due to rounding.
  • For example, $7^2$ needs parentheses for accurate entry.

Rationalizing and Simplifying Limits

• Rationalization Techniques
  • Use conjugates to simplify limits involving square roots, especially in cases of indeterminate forms.

Special Limit Theorems

• Squeeze Theorem
  • If a function f(x) is trapped between functions g(x) and h(x) that approach the same limit, then $\lim_{x \to c} f(x)$ also approaches that limit.
• Notable Limits
  • $\lim_{x \to 0} \frac{\sin(x)}{x} = 1$ and
  • $\lim_{x \to 0} \frac{1 - \cos(x)}{x} = 0$
  • These notable limits help resolve many limit problems analytically.

Trigonometric Functions and Limits

• Trigonometric Functions
  • Most trig functions like sine and cosine are defined for all real numbers, but tangent, cosecant, secant and cotangent have restrictions based on undefined points (division by zero).
• Order of Operations in Trig
  • Important to know how the values of sine, cosine, and tangent behave near their discontinuities.

Conclusion

• Methods to Solve Limits
  • Analytical methods including simplification, numerical evaluations, graphical approaches, and recognition of specific limits.
• Importance of Understanding Various Approaches
  • Knowing when to apply which method facilitates resolution of limit problems effectively and accurately.