Calc-1 Discussion 03

Discussion Topics covered: Evaluating limits of composite functions, directional and asymptotic boundaries, special trigonometric limits, Squeeze Theorem on bounded oscillations, the three-part continuity rubric, classifying discontinuities, solving for continuous piecewise parameters, binomial expansion, combining rational expressions and complex fractions, negative and rational exponents

Part A: Review This Week's Lecture Topics (~30 mins)

Evaluating Limits of Composite Functions ("Inside-Out" Evaluation)

• Evaluating limits of nested functions $\lim_{x \to c} f(g(x))$ using sequential "inside-out" evaluation, tracking the intermediate state $u = g(x)$ and its directional target
• Applying the outer continuity rule $\lim_{x \to c} f(g(x)) = f(\lim_{x \to c} g(x))$ when the outer function is continuous at the inner limit value
• Evaluating composite limits with infinite and directional boundaries into transcendental functions (e.g., $\lim_{x \to 0^+} \tan^{-1}(\ln x) = -\frac{\pi}{2}$ and $\lim_{x \to 0^-} e^{1/x} = 0$)
• Resolving indeterminate forms at infinity for composite exponential quotients by dividing numerator and denominator by the dominant exponential term (e.g., $\lim_{x \to \infty} \frac{1 + e^x}{1 - e^x} = -1$)

Special Trigonometric Limits & The Squeeze Theorem

• Applying fundamental geometric limit identities to resolve $\frac{0}{0}$ trigonometric indeterminate forms: $\lim_{\theta \to 0} \frac{\sin(\theta)}{\theta} = 1$ and $\lim_{\theta \to 0} \frac{1 - \cos(\theta)}{\theta} = 0$
• Scaling constant coefficients and matching algebraic arguments in trigonometric ratios (e.g., evaluating $\lim_{x \to 0} \frac{\sin(ax)}{bx} = \frac{a}{b}$ and splitting tangents/sines like $\lim_{x \to 0} \frac{\tan(3x)}{\sin(7x)} = \frac{3}{7}$)
• Applying the Squeeze (Sandwich) Theorem by bounding oscillating terms between solvable algebraic envelopes (e.g., $-x^2 \le x^2 \sin(1/x) \le x^2$ to prove $\lim_{x \to 0} x^2 \sin(1/x) = 0$)
• Setting up the formal two-part proof requirement for the Squeeze Theorem: explicitly stating the bounding inequality and proving both outer functions converge to the identical finite limit

Continuity Verification, Classification, and Piecewise Parameters

• Proving continuity at a point $x = c$ using the rigorous Three-Part Continuity Rubric:
  1. $f(c)$ is defined
  2. $\lim_{x \to c} f(x)$ exists ($\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) = L$)
  3. $\lim_{x \to c} f(x) = f(c)$
• Diagnosing and classifying discontinuities across piecewise and rational functions:
  • Removable Discontinuity (Hole): $\lim_{x \to c} f(x)$ exists, but $f(c)$ is undefined or $f(c) \neq \lim_{x \to c} f(x)$
  • Jump Discontinuity (Step): Left- and right-hand limits are finite but unequal, with jump magnitude $\Delta y = \left|\lim_{x \to c^+} f(x) - \lim_{x \to c^-} f(x)\right|$
  • Infinite Discontinuity (Vertical Asymptote): One or both one-sided limits blow up to $\pm\infty$
• Enforcing global continuity across piecewise boundaries by equating one-sided limits and defined function values to solve for unknown parameter constants (e.g., finding $a$, $b$, or $k$)
• Constructing or interpreting coordinate sketches and asymptote equations from multi-point piecewise limit and continuity specifications

Part B: Next Week's Prerequisite Refresher (~30 mins)

Review the following prerequisite skills:

• Binomial Expansion
• Combining Rational Expressions and Complex Fractions
• Negative and Rational Exponents

Part C: General Q/A

Open the floor to questions from students regarding any topic