Comprehensive Calculus Notes on Limits, Asymptotes, Rational Behavior, and the Squeeze Theorem

Vertical Asymptotes and One-Sided Infinite Limits

• Input and Output Coordinate Concepts:

  • The location along the independent variable axis is represented by the input $x$-coordinate (referred to as the $x$-valve coordinate).
  • The corresponding function output value is represented by the $y$-coordinate (referred to as the output valve $y$-coordinate).
  • Example: For a 1-tank system modeling \n\sin(x)\n, the derivative relationship yields \n\cos(x)\n

• One-Sided Limits of Trigonometric Functions:

  • Approaching \n\frac{\pi}{2}\n from the left for the tangent function yields positive infinity: \n\lim_{x \to \frac{\pi}{2}^-} \tan(x) = +\infty\n
  • Approaching \n\frac{\frac{\pi}{2}}\n from the right for the tangent function yields negative infinity: \n\lim_{x \to \frac{\pi}{2}^+} \tan(x) = -\infty\n

• Definition of Vertical Asymptotes:

  • If a function yields an infinite limit (either $+\infty$ or $-\infty$) as $x$ approaches a constant $a$ from either the left or the right side, then the line $x = a$ is a vertical asymptote for the graph of the function.

• Non-Existence of Limits at Asymptotes:

  • When the left-hand limit and right-hand limit approach opposite infinities (e.g., $+\infty$ on one side and $-\infty$ on the other side), the two-sided limit does not exist (DNE).

Limits at Infinity and Inverse Function Behavior

• Graphical Interpretation of Limits at Infinity:

  • The limit expression \n\lim_{x \to +\infty} f(x)\n describes the asymptotic end behavior of the function on the far right side of the graph.
  • The limit expression \n\lim_{x \to -\infty} f(x)\n describes the asymptotic end behavior of the function on the far left side of the graph.

• Reflective Geometry of Inverse Functions:

  • The graph of a function $f(x)$ and the graph of its inverse function $f^{-1}(x)$ are geometric reflections of each other across the line $y = x$.

• Asymptotic Relationship Between Tangent and Arctangent:

  • The function $y = \tan(x)$ possesses vertical asymptotes at $x = \frac{\pi}{2}$ and $x = -\frac{\pi}{2}$.
  • Reflecting these vertical lines across $y = x$ causes the inverse function $y = \arctan(x)$ (also denoted as $y = \tan^{-1}(x)$) to possess horizontal asymptotes at $y = \frac{\pi}{2}$ and $y = -\frac{\pi}{2}$.
  • Evaluating limits at positive and negative infinity for the inverse tangent function yields: \n\lim_{x \to +\infty} \arctan(x) = \frac{\pi}{2}\n \n\lim_{x \to -\infty} \arctan(x) = -\frac{\pi}{2}\n

End Behavior of Rational and Exponential Functions

• Asymmetric Behavior of Exponential Functions:

  • Exponential functions behave differently as $x$ approaches positive infinity versus negative infinity: \n\lim_{x \to +\infty} e^x = +\infty\n \n\lim_{x \to -\infty} e^x = 0\n

• Degree Rules for Rational Functions:

  • For a rational function defined as $f(x) = \frac{\text{polynomial}}{\text{polynomial}}$, end behavior as $x \to \pm \infty$ is dictated by comparing the polynomial degrees of the numerator and denominator:
  • Rule 1 (Numerator Degree > Denominator Degree): If the degree of the numerator is strictly greater than the degree of the denominator, then: \n\lim_{x \to \pm \infty} f(x) = \pm \infty\n
  • Rule 2 (Denominator Degree > Numerator Degree): If the degree of the denominator is strictly greater than the degree of the numerator, then: \n\lim_{x \to \pm \infty} f(x) = 0\n
  • Rule 3 (Equal Degrees): If the degree of the numerator is equal to the degree of the denominator, then the limit as $x \to \pm \infty$ equals the ratio of the leading coefficients: \n\lim_{x \to \pm \infty} f(x) = \text{ratio of leading coefficients}\n

• Algebraic Method for Rational Limits at Infinity:

  • An alternative algebraic technique to evaluate rational function limits at infinity is to take the original function and divide every term in both the numerator and denominator by the highest power of $x$ present in the denominator.

• Worked Rational Function Examples:

  • Equal Degrees Example: Evaluate \n\lim_{x \to +\infty} \frac{3x^2 - 2x + 1}{5x^2 - 4x + 5}\n
    • The degree of the numerator is $2$ and the degree of the denominator is $2$.
    • Because the degrees are equal, the limit is the ratio of the leading coefficients \n\frac{3}{5}\n: \n\lim_{x \to +\infty} \frac{3x^2 - 2x + 1}{5x^2 - 4x + 5} = \frac{3}{5}\n
  • Higher Denominator Degree Example: Evaluate \n\lim_{x \to +\infty} \frac{3x^2 - 2x + 1}{5x^3 - 3x + 11}\n
    • The degree of the numerator is $2$ and the degree of the denominator is $3$.
    • Because the denominator degree is strictly larger, the limit is $0$: \n\lim_{x \to +\infty} \frac{3x^2 - 2x + 1}{5x^3 - 3x + 11} = 0\n
  • Higher Numerator Degree Example: Evaluate \n\lim_{x \to -\infty} \frac{3x^3 - 2x + 1}{5x^2 - 4x + 3}\n
    • The numerator degree $3$ exceeds the denominator degree $2$, driving the limit to $-\infty$.

Techniques for Evaluating Limits at Negative Infinity

• Variable Transformation Rule:

  • To evaluate a limit as $x \to -\infty$ for a general function $f(x)$, transform the variable to positive infinity using the equivalence: \n\lim_{x \to -\infty} f(x) = \lim_{x \to +\infty} f(-x)\n

• Exponential Quotient Limit at Positive Infinity:

  • Evaluate \n\lim_{x \to +\infty} \frac{e^x}{e^x + 4}\n
  • Divide numerator and denominator by $e^x$: \n\lim_{x \to +\infty} \frac{1}{1 + \frac{4}{e^x}}\n
  • Since \n\lim_{x \to +\infty} \frac{4}{e^x} = 0\n, the expression simplifies to: \n\frac{1}{1 + 0} = 1\n

• Exponential Quotient Limit at Negative Infinity:

  • Evaluate \n\lim_{x \to -\infty} \frac{e^x}{e^x + 4}\n
  • Apply the transformation rule \n\lim_{x \to +\infty} f(-x)\n: \n\lim_{x \to +\infty} \frac{e^{-x}}{e^{-x} + 4} = \lim_{x \to +\infty} \frac{\frac{1}{e^x}}{\frac{1}{e^x} + 4}\n
  • Since \n\lim_{x \to +\infty} \frac{1}{e^x} = 0\n, evaluating the expression yields: \n\frac{0}{0 + 4} = 0\n

• Radical Rational Function Limits:

  • Consider the function \nf(x) = \frac{x^3 - 1}{\sqrt{x^6 + 1}}\n
  • For positive infinity, noting that \n\sqrt{x^6} = x^3\n for $x > 0$: \n\lim_{x \to +\infty} \frac{x^3 - 1}{\sqrt{x^6 + 1}} = 1\n
  • For negative infinity, noting that \n\sqrt{x^6} = |x^3| = -x^3\n for $x < 0$: \n\lim_{x \to -\infty} \frac{x^3 - 1}{\sqrt{x^6 + 1}} = -1\n

Squeeze Theorem for Infinite and Ill-Behaved Limits

• Formal Statement of the Squeeze Theorem:

  • Suppose that \ng(x) \le f(x) \le h(x)\n for all $x$ near a point $a$ (except possibly at $a$ itself).
  • Suppose further that the outer limits exist and are equal: \n\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L\n
  • Then the limit of the inner function $f(x)$ is trapped and forced to equal the same value: \n\lim_{x \to a} f(x) = L\n

• Application to Oscillatory Limits at Infinity:

  • Evaluate the limit of an ill-behaved oscillating function at infinity: \n\lim_{x \to +\infty} \frac{\sin(x)}{x}\n
  • Step 1: Establish known bounds. The sine function is strictly bounded for all real numbers: \n-1 \le \sin(x) \le 1\n
  • Step 2: Construct the inequality. Divide all terms in the inequality by $x$ (assuming $x > 0$ as $x \to +\infty$): \n\frac{-1}{x} \le \frac{\sin(x)}{x} \le \frac{1}{x}\n
  • Step 3: Evaluate the bounding limits. Take the limit of the bounding functions as $x \to +\infty$: \n\lim_{x \to +\infty} \left(\frac{-1}{x}\right) = 0\n \n\lim_{x \to +\infty} \left(\frac{1}{x}\right) = 0\n
  • Step 4: Conclude via Squeeze Theorem. Because \n\frac{\sin(x)}{x}\n is stuck between two bounding functions that both approach $0$ as $x \to +\infty$, by the Squeeze Theorem: \n\lim_{x \to +\infty} \frac{\sin(x)}{x} = 0\n

Hierarchy of Function Growth Rates

• Growth Rate Ordering as $x \to +\infty$:
  • Different families of functions grow towards infinity at fundamentally different rates.
  • Exponential functions grow faster than polynomial functions.
  • Polynomial functions grow faster than logarithmic functions.
  • Complete dominance hierarchy from fastest-growing to slowest-growing: \n\text{Exponential Functions} > \text{Polynomial Functions} > \text{Logarithmic Functions}\n