Calculus: Limits at Infinity, Indeterminate Forms, and Graphical Transformations

Redefining Functions and Removable Discontinuities

• Expression evaluation and point redefinition:
  • Given the rational expression $\frac{x+1}{x-3}$, plugging in $x = 3$ results in division by zero.
  • Evaluating the numerator at $x = 3$ yields $3 + 1 = 4$
  • To address the discontinuity at $x = 3$, the function must be redefined at that point such that $f(3) = 4$

Limits at Infinity and Relative Growth Rates

• Comparison of Power Functions vs. Exponential Functions:
  • Consider the ratio of a power function to an exponential function as $x \to \infty$: $\frac{100^4}{4^{100}}$
  • The numerator $100^4$ is a fixed large number, but the denominator $4^{100}$ represents $4 \times 4 \times 4 \dots$ multiplied $100$ times, making the denominator significantly larger than the numerator.
  • Extending the inputs to larger numbers such as $\frac{200^4}{4^{200}}$, the top is $200^4$ while the bottom is $4^{200}$ ($4$ multiplied $200$ times).
  • General Limit Behavior: For any base $a > 1$ and power function exponent, the exponential function grows vastly faster than the power function:     $\lim_{x \to \infty} \frac{x^k}{a^x} = 0$
  • Dividing a power term by a much larger exponential term causes the expression to approach $0$

Dominant Term Method for Indeterminate Forms

• Evaluating Indeterminate Form $\frac{\infty}{\infty}$:
  • Definition of Dominant Term: When evaluating limits as $x \to \infty$ that yield $\frac{\infty}{\infty}$, isolate the highest power term (dominant term) in the numerator and denominator and discard all lower-degree terms.
  • Shortened Evaluation Technique:
  • Retain only the term with the largest exponent in both numerator and denominator.
  • Simplify the resulting fraction before taking the limit.
  • Infinite Limits vs. Does Not Exist (DNE):
  • For an example simplification yielding $\frac{x^4}{x^3} = x$, as $x \to \infty$, the limit approaches $\infty$
  • Traditional Limit Definition: A limit is strictly defined as a finite number. Because $\infty$ is not a finite number, some conventions designate limits evaluating to $\infty$ as $\text{DNE}$ (Does Not Exist), though writing $\infty$ describes the precise unbounded behavior.
  • Horizontal Asymptotes:
  • When the limit of $y = f(x)$ as $x \to \infty$ or $x \to -\infty$ approaches a finite constant $c$, the line $y = c$ represents a horizontal asymptote.
  • Example: If $y$ gets arbitrarily close to $3$ as $x \to \infty$, the horizontal asymptote is $y = 3$

Graphical Analysis of Limits Using Desmos

• Plotting Functions in Desmos:
  • Input expression format: $f(x) = \sqrt{x^2 + x + 5}$
  • Key command for square root: Use SQRT syntax, such as SQRT(x^2 + x + 5).
  • Adjusting the viewing window:
  • Open the graph window settings to configure the $x$-window range and $y$-range.
  • Set bounds (e.g., $x$ from $-100$ to $-5$) to observe end-behavior as $x \to -\infty$
• Numerical Approximations via Table Feature:
  • Use the built-in Desmos table feature to evaluate function values near target limits.
  • Observe intermediate outputs (e.g., values near $2.035$) to accurately guess or confirm limit values.

Algebraic Rationalization for Indeterminate Form Infinity Minus Infinity

• The Indeterminate Form $\infty - \infty$:
  • $\infty - \infty$ is an indeterminate form and CANNOT be assumed to equal $0$
  • As variables expand into larger domains ("larger boxes"), one infinite quantity may grow significantly faster than another, or both terms may be equivalent in order of magnitude.
• Conjugate Pair Rationalization Method:
  • Used to evaluate indeterminate expressions involving square roots.
  • Step 1: Identify radical expression in the form $a - b$
  • Step 2: Multiply and divide by the conjugate pair $(a + b)$:     $\frac{(a - b)(a + b)}{a + b} = \frac{a^2 - b^2}{a + b}$
  • Step 3: Expand the numerator using difference of squares to remove the square root: $a^2 - b^2$
  • Step 4: Retain the sum $a + b$ in the denominator.
  • Step 5: Combine like terms in the numerator (e.g., collecting terms like $-6x^2$ and $-2$) and isolate dominant terms on the top and bottom to evaluate the limit.

Function Transformations and Point-by-Point Table Analysis

• Coordinate Table Analysis for Graph Transformations:
  • Graph transformations can be analyzed point-by-point using coordinate tracking tables.
  • Map original $(x, y)$ points from the primary function to transformed $(x, y)$ points on the new function.
• Transformation Examples and Point Shifts:
  • Original Point 1: $(-4, 2)$
  • Original Point 2: $(5, 4)$
  • Transformation Rules Applied:
  • Horizontal translations modify the $x$-row values (e.g., shifting $x = 5$ to $x = 6$).
  • Vertical scaling modifies the $y$-row values (e.g., dividing the original $y$-value $4$ by $2$ yields $1$).
  • Shifted Point Result: $(5, 4)$ transforms to $(6, 1)$.
  • Horizontal asymptote lines and boundary points shift correspondingly across the transformed domain.