Comprehensive Calculus Guide: Limits, One-Sided Behavior, and Asymptotes

Mathematical Foundation of Limits

• A limit provides a mathematically precise framework for analyzing the behavior of a function near a specific point, regardless of the function's value at that exact point.
• Intuitive concepts in mathematics are often formalized into logically precise definitions, which can increase structural and notation complexity.
• Initial numerical exploration example:
  • Function inputs evaluated near $x = 1$: $x = 0$, $x = \frac{1}{2}$, $x = 0.9$, and $x = 0.99$.
  • As $x$ approaches $1$, the output values $f(x)$ approach $2$.

Comparative Analysis of Limits vs. Function Values

To demonstrate the distinct behavior of limits versus actual function values at a point $x = 2$, consider three functions $f(x)$, $g(x)$, and $h(x)$ that are identical everywhere except at $x = 2$:

• Function $f(x)$:

  • Behavior: Continuous path passing through $(2, 3)$.
  • Limit evaluation: $\lim_{x \to 2} f(x) = 3$.
  • Function value: $f(2) = 3$.
  • Relationship: The limit and the function value agree ($\lim_{x \to 2} f(x) = f(2)$).

• Function $g(x)$:

  • Behavior: Has an open circle at $(2, 3)$ with no assigned point elsewhere.
  • Limit evaluation: $\lim_{x \to 2} g(x) = 3$.
  • Function value: $g(2)$ is undefined (does not exist).
  • Relationship: The limit exists, but the function value does not exist.

• Function $h(x)$:

  • Behavior: Has an open circle at $(2, 3)$ and a solid point at $(2, 1)$.
  • Limit evaluation: $\lim_{x \to 2} h(x) = 3$.
  • Function value: $h(2) = 1$.
  • Relationship: Both the limit and the function value exist, but they are not equal ($\lim_{x \to 2} h(x) \neq h(2)$).

• Relationship to Continuity:

  • The condition where the limit of a function equals the function's actual value, $\lim_{x \to a} f(x) = f(a)$, defines the concept of continuity at $x = a$.

Pitfalls of Numerical Sampling with Oscillating Functions

Evaluating limits solely by plugging in numbers can lead to incorrect conclusions if sampling coincides with specific repeating values.

• Example Function: $f(x) = \sin\left(\frac{\pi}{x}\right)$ as $x \to 0$.
• Misleading Numerical Sampling Table:
  • For $x = \frac{1}{2}$: $f\left(\frac{1}{2}\right) = \sin(2\pi) = 0$
  • For $x = \frac{1}{10}$: $f\left(\frac{1}{10}\right) = \sin(10\pi) = 0$
  • For $x = \frac{1}{100}$: $f\left(\frac{1}{100}\right) = \sin(100\pi) = 0$
  • Erroneous conclusion from table: Numerical evaluation suggests $\lim_{x \to 0} \sin\left(\frac{\pi}{x}\right) = 0$.
• Correct Analysis:
  • Evaluating at $x = \frac{2}{5}$ gives $f\left(\frac{2}{5}\right) = \sin\left(\frac{5\pi}{2}\right) = 1$.
  • In general, odd integer multiples of $\frac{\pi}{2}$ yield function values of $1$ or $-1$.
  • Graph behavior: As $x$ approaches $0$, the function oscillates infinitely fast between $-1$ and $1$.
  • Within any arbitrary interval around $0$, the function takes on every value from $-1$ to $1$ infinitely many times.
  • Conclusion: Because no single value is being approached, $\lim_{x \to 0} \sin\left(\frac{\pi}{x}\right)$ does not exist (DNE).

One-Sided Limits and Fundamental Limit Theorem

• Definitions:

  • Right-hand limit: Approaching $a$ from the right where $x > a$, denoted as $\lim_{x \to a^+} f(x)$.
  • Left-hand limit: Approaching $a$ from the left where $x < a$, denoted as $\lim_{x \to a^-} f(x)$.

• Theorem:

  • $\lim_{x \to a} f(x) = L$ if and only if $\lim_{x \to a^+} f(x) = L$ and $\lim_{x \to a^-} f(x) = L$.
  • For a two-sided limit to exist, both one-sided limits must exist and be strictly equal.

• Graphical Piecewise Example:

  • At $x = -2$:
  • $\lim_{x \to -2^-} f(x) = 0$
  • $\lim_{x \to -2^+} f(x) = 0$
  • $\lim_{x \to -2} f(x) = 0$
  • $f(-2)$ is undefined (open circle).
  • At $x = 1$:
  • Left-hand limit: $\lim_{x \to 1^-} f(x) = -4$
  • Right-hand limit: $\lim_{x \to 1^+} f(x) = 1$
  • Two-sided limit: $\lim_{x \to 1} f(x) = \text{DNE}$ (does not exist) because $\lim_{x \to 1^-} f(x) \neq \lim_{x \to 1^+} f(x)$.
  • Function value: $f(1) = -4$ (solid circle at $(1, -4)$).

Infinite Limits and Vertical Asymptotes

• Modes of Limit Non-Existence:

  1. Infinite oscillation near a point (e.g., $\sin\left(\frac{\pi}{x}\right)$).
  2. Disagreement between left-hand and right-hand limits (break/jump discontinuities).
  3. Unbounded growth where output values become arbitrarily large positive or negative values.

• Infinite Limit Notation and Interpretation:

  • If $f(x)$ grows arbitrarily large positively as $x \to a$, write $\lim_{x \to a} f(x) = \infty$.
  • If $f(x)$ grows arbitrarily large negatively as $x \to a$, write $\lim_{x \to a} f(x) = -\infty$.
  • Critical distinction: Writing $\lim_{x \to a} f(x) = \infty$ describes a specific, structured way in which the limit fails to exist; the limit still technically does not exist in the real number system.

• Vertical Asymptotes:

  • Infinite limits correspond directly to vertical asymptotes on a graph.

• Graph Analysis with Vertical Asymptotes at $x = -1$ and $x = 2$:

  • At $x = -1$:
  • $\lim_{x \to -1^-} f(x) = \infty$
  • $\lim_{x \to -1^+} f(x) = \infty$
  • $\lim_{x \to -1} f(x) = \infty$
  • At $x = 2$:
  • $\lim_{x \to 2^-} f(x) = -\infty$
  • $\lim_{x \to 2^+} f(x) = \infty$
  • $\lim_{x \to 2} f(x) = \text{DNE}$ (does not exist, as left and right tendencies disagree and do not yield a uniform infinite sign).

Analytical and Graphical Analysis of Rational Functions

• Evaluating $\lim_{x \to 1^+} \frac{1}{x - 1}$ numerically:

  • Select $x = 1.1$: $\frac{1}{1.1 - 1} = \frac{1}{\frac{1}{10}} = 10$
  • Select $x = 1.01$: $\frac{1}{1.01 - 1} = \frac{1}{\frac{1}{100}} = 100$
  • Select $x = 1.001$: $\frac{1}{1.001 - 1} = \frac{1}{\frac{1}{1000}} = 1000$
  • Result: $\lim_{x \to 1^+} \frac{1}{x - 1} = \infty$.

• Evaluating $\lim_{x \to 1^-} \frac{1}{x - 1}$ numerically:

  • Select $x = 0.9$: $\frac{1}{0.9 - 1} = \frac{1}{-\frac{1}{10}} = -10$
  • Select $x = 0.99$: $\frac{1}{0.99 - 1} = \frac{1}{-\frac{1}{100}} = -100$
  • Result: $\lim_{x \to 1^-} \frac{1}{x - 1} = -\infty$.

• Graphical Transformations:

  • The parent function $g(x) = \frac{1}{x}$ has a vertical asymptote at x = 0$.\n - Replacing x$with$x - 1$in$f(x) = \frac{1}{x - 1}$represents a horizontal shift to the right by$1 unit.\n - Consequently, the vertical asymptote shifts from x = 0$to$x = 1$$.