Introduction to Limits and Limit Properties

Introduction to Limits and Notation

• Definitions and Fundamental Concepts:
  • A limit is defined as a $y$ value approached by a function as the $x$ value used in the function approaches a fixed value of $c$.
  • The fundamental principle of a limit is that it represents an approached value, not necessarily the value of the function at that exact point.
• Notation:
  • Standard notation for a limit uses a script abbreviation: $\lim_{x \to c} f(x)$.
  • This expression is read as "the limit as $x$ approaches $c$ of $f(x)$."
• Conditions for Existence:
  • For a limit to exist, the function must approach the same $y$ value from both the left side and the right side of the constant $c$.
  • If the paths from either side do not meet at the same location, the limit does not exist (DNE).

Graphical Analysis of Limits

• Function Verification:
  • Before evaluating limits graphically, verify the relation is a function using the vertical line test. A relation is a function if every vertical line intersects the graph no more than once.
• Comparative Analysis of Limits and Function Values:
  • Example Case 1: At $x = -6$, the function approaches the same location from both sides. The limit is $3$.
  • Example Case 2: At $x = -3$, the paths from the left and right do not meet. Consequently, $\lim_{x \to -3} f(x)$ does not exist. However, the actual function value exists, defined as $f(-3) = 1$.
  • Example Case 3: At $x = 0$, both sides of the graph converge at a single point on the $y$-axis, yielding a limit of $-2$.
  • Example Case 4: At $x = 2$, there is a hole in the graph. The limit as $x$ approaches $2$ is $-1$, even though the point is not there. The actual function value is located elsewhere, such as at a solid dot at $f(2) = 3$.
  • Example Case 5: At $x = 5$, the sides approach different values (the left at $2$ and the right at $5$). In this case, the limit does not exist.
• Common Reasons for Non-Existence:
  • The primary reason a limit fails to exist is that the two sides of the function do not approach the same value.
  • Specificity can be added by noting that one side shoots toward positive infinity while the other shoots toward negative infinity, or that they simply approach distinct finite numbers.

Infinite Limits and Asymptotes

• Vertical Asymptotes:
  • Consider the function $g(x) = \frac{1}{x^2}$. As $x$ approaches $0$, the graph shoots upward infinitely on both sides.
  • While some perspectives classify this as "does not exist" due to the lack of a finite value, it is standard to identify the limit as $\infty$ when both sides approach that same direction within the context of calculus.
  • For the reflected function $h(x) = -\frac{1}{x^2}$, as $x$ approaches $0$, both sides shoot downward, meaning the limit is $-\text{infinity}$.

Numerical Estimation via Tabular Approximation

• Methodology:
  • When a graph is unavailable and algebraic manipulation is unknown, a tabular or table approximation can be used to estimate a limit.
  • This involves selecting $x$ values increasingly close to the target value $c$ from both the left and the right sides.
• Example: Estimating the limit as $x$ approaches $2$:
  • From the left:
    • At $x = 1.8$, $f(x) \text{ is } 0.67214$.
    • At $x = 1.9$, $f(x) \text{ is } 0.57216$.
    • At $x = 1.99$, $f(x) \text{ is } 0.50633$.
    • At $x = 1.999$, $f(x) \text{ is } 0.50063$.
  • From the right:
    • At $x = 2.2$, $f(x) \text{ is } 0.40056$.
    • At $x = 2.1$, $f(x) \text{ is } 0.4446$.
    • At $x = 2.01$, $f(x) \text{ is } 0.49381$.
    • At $x = 2.001$, $f(x) \text{ is } 0.49938$.
  • Convergence: Since both sides appear to converge toward a specific decimal, the estimated limit is approximately $0.5$.
• Example: Estimating the limit as $x$ approaches $-1$ for $\frac{x^3+1}{5x+5}$:
  • Using values like $-1.2$, $-1.1$, and $-1.01$, the results show an approach toward values around $1.66$ or $1.67$.

The Formal Epsilon-Delta Definition of a Limit

• Definition Statement:
  • Let $L$ be a finite number, and let $f(x)$ be defined on an open interval containing the value $c$.
  • The statement $\lim_{x \to c} f(x) = L$ means that for any $\epsilon > 0$, there exists a $\delta > 0$ such that if $|x - c| < \rho$, then $|f(x) - L| < \rho$.
• Conceptual Meaning:
  • $\epsilon$ (epsilon) represents a chosen vertical distance or interval around the limit value $L$.
  • $\delta$ (delta) represents the required horizontal distance or interval around the target value $c$ needed to guarantee that every function value falls within the $\epsilon$ interval of $L$.
  • If one can specify a $\delta$ for any given $\epsilon$, the limit is proven to be $L$.
• Computational Examples for $\delta$:
  • Linear Case: Given $f(x) = 6x + 4$ and $c = 5$, find the required $\delta$ for $\epsilon = 0.001$.
    1. Determine $L$ by evaluating $f(5) = 6(5) + 4 = 34$.
    2. Set up the inequality $|f(x) - L| < 0.001$, leading to $|6x + 4 - 34| < 0.001$.
    3. Simplify to $|6x - 30| < 0.001$, then factor out the constant to get $6|x - 5| < 0.001$.
    4. Divide by the constant to isolate the $x-c$ form: $|x - 5| < \frac{0.001}{6}$.
    5. Therefore, $\delta = \frac{0.001}{6}$.
  • Variable Case: Given $f(x) = 8 - \frac{1}{3}x$ and $c = 24$, find $\delta$ in terms of $\epsilon$ where $L = 0$.
    1. Set up $|8 - \frac{1}{3}x - 0| < \rho$.
    2. Factor out the coefficient to isolate $x$: $|-\frac{1}{3}(x - 24)| < \rho$.
    3. Move the scalar outside the absolute value sign: $\frac{1}{3}|x - 24| < \rho$.
    4. Multiply by $3$ to find the delta form: $|x - 24| < 3\rho$.
    5. Therefore, $\delta = 3\rho$.

Algebraic Properties of Limits

• Foundational Laws (where $k$ is a constant):
  1. Constant Law: $\lim_{x \to c} k = k$.
  2. Identity Law: $\lim_{x \to c} x = c$.
  3. Scalar Multiple Law: $\lim_{x \to c} [k \times f(x)] = k \times \text{lim}_{x \to c} f(x)$.
  4. Sum/Difference Law: $\lim_{x \to c} [f(x) \text{ plus or minus } g(x)] = \text{lim}_{x \to c} f(x) \text{ plus or minus } \text{lim}_{x \to c} g(x)$.
  5. Product Law: $\lim_{x \to c} [f(x) \times g(x)] = [\text{lim}_{x \to c} f(x)] \times [\text{lim}_{x \to c} g(x)]$.
  6. Quotient Law: $\lim_{x \to c} \frac{f(x)}{g(x)} = \frac{\text{lim}_{x \to c} f(x)}{\text{lim}_{x \to c} g(x)}$, provided the denominator limit is non-zero.
  7. Power Law: $\lim_{x \to c} [f(x)]^n = [\text{lim}_{x \to c} f(x)]^n$ for integer powers.
  8. Composition Law: $\lim_{x \to c} f(g(x)) = f(\text{lim}_{x \to c} g(x))$.

Evaluation Examples Using Limit Properties

• Multi-Step Evaluation:
  • Given $\lim_{x \to 4} f(x) = -3$ and $\lim_{x \to 4} g(x) = 5$, evaluate $\lim_{x \to 4} [6 f(x) + g(x)^2 + x]$.
    1. Distribute limits over the sums: $\lim 6 f(x) + \text{lim } g(x)^2 + \text{lim } x$.
    2. Pull constants and use power rules: $6 \times (\text{lim } f(x)) + (\text{lim } g(x))^2 + \text{lim } x$.
    3. Substitute known values: $6(-3) + (5)^2 + 4$.
    4. Calculation: $-18 + 25 + 4 = 11$.
• Rational Function Evaluation:
  • Evaluate $\lim_{x \to 4} \frac{2g(x) + 13}{f(x)g(x) - x}$.
    1. Apply the quotient rule to separate top and bottom limits.
    2. Apply the scalar and product rules: $\frac{2 \times \text{lim } g(x) + 13}{(\text{lim } f(x) \times \text{lim } g(x)) - \text{lim } x}$.
    3. Substitute values: $\frac{2(5) + 13}{(-3)(5) - 4}$.
    4. Calculation: $\frac{23}{-19}$.
• Evaluation via Direct Substitution:
  • For continuous functions, properties allow for the direct substitution of the approach value $c$ for every occurrence of $x$.
  • Polynomial example: $\lim_{x \to -5} (7x - x^3) = 7(-5) - (-5)^3 = -35 + 125 = 90$.
  • Square root example: $\lim_{x \to -3} \text{sqrt}(x^2 + 12x + 29) = \text{sqrt}((-3)^2 + 12(-3) + 29) = \text{sqrt}(9 - 36 + 29) = \text{sqrt}(2)$.
  • Trigonometric example: $\lim_{x \to \frac{\text{pi}}{6}} \text{sin}(2x) = \text{sin}(2 \times \frac{\text{pi}}{6}) = \text{sin}(\frac{\text{pi}}{3}) = \frac{\text{sqrt}(3)}{2}$.
  • Natural log example: Evaluation of a function results in the simplified form $\ln(104 - 100) = \text{ln}(4)$.