Continuous Functions: Sign Analysis, Domains, Limits, and Indeterminate Forms

Sign of a Function

First-Degree Polynomials

A first-degree polynomial is defined by the general linear expression $f(x) = ax + b$.

To find the root (zero) of the function, set $f(x) = 0$: $f(x) = 0 \implies x = -\frac{b}{a}$

Sign Distribution Table

• For $x \in \left(-\infty, -\frac{b}{a}\right)$: The sign of $f(x)$ is opposite to the sign of $a$.
• At $x = -\frac{b}{a}$: $f(x) = 0$
• For $x \in \left(-\frac{b}{a}, +\infty\right)$: The sign of $f(x)$ is the same as the sign of $a$.

Worked Example

Given the polynomial $f(x) = 2x - 4$: $f(x) = 0$$2x - 4 = 0$$2x = 4$$x = \frac{4}{2} = 2$

Second-Degree Polynomials

A second-degree polynomial is defined by the general quadratic formula $f(x) = ax^2 + bx + c$.

The discriminant $\Delta$ determines the existence and nature of the real roots: $\Delta = b^2 - 4ac$

1st Case: $\Delta > 0$

When $\Delta > 0$, the function has two distinct real roots: $x' = \frac{-b - \sqrt{\Delta}}{2a}$$x'' = \frac{-b + \sqrt{\Delta}}{2a}$

Sign distribution along the real number line:

• For $x \in (-\infty, x')$: Same sign as $a$
• At $x = x'$: $f(x) = 0$
• For $x \in (x', x'')$: Opposite sign to $a$
• At $x = x''$: $f(x) = 0$
• For $x \in (x'', +\infty)$: Same sign as $a$

2nd Case: $\Delta = 0$

When $\Delta = 0$, the function has a single double root: $x' = x'' = -\frac{b}{2a}$

Sign distribution along the real number line:

• For $x \in (-\infty, x')$: Same sign as $a$
• At $x = x'$: $f(x) = 0$
• For $x \in (x', +\infty)$: Same sign as $a$

3rd Case: $\Delta < 0$

When $\Delta < 0$, there are no real roots.

Sign distribution along the real number line:

• For all $x \in (-\infty, +\infty)$: The sign of $f(x)$ is always the same as the sign of $a$.

Shortcuts for Finding Quadratic Roots

Specific relationships between coefficients allow immediate determination of the roots without explicitly evaluating $\Delta$:

• If $a + b + c = 0$, then the roots are $x' = 1$ and $x'' = \frac{c}{a}$.
• If $a - b + c = 0$, then the roots are $x' = -1$ and $x'' = -\frac{c}{a}$.

Application Example for Second-Degree Polynomials

Consider the function $f(x) = x^2 - 3x + 2$.

• Scientific calculator sequence: mode 5 -> 3
• Using coefficient shortcut: Here $a = 1$, $b = -3$, and $c = 2$. Since $a + b + c = 1 - 3 + 2 = 0$, the roots are:   $x' = 2$$x'' = 1$
• Since $\Delta > 0$ and $a = 1 > 0$:
  • For $x \in (-\infty, 1)$: $f(x) > 0$ ($+$)
  • At $x = 1$: $f(x) = 0$
  • For $x \in (1, 2)$: $f(x) < 0$ (-$)\n - At x = 2$:$f(x) = 0\n - For x \in (2, +\infty)$:$f(x) > 0$($+)\n\n\n# Domain of Definition\n\n## Rules for Function Types\n\n- **Polynomial Functions**: The domain of definition is all real numbers, \mathbb{R}$or$]-\infty, +\infty[.\n- **Rational Functions**: For a function f(x) = \frac{A(x)}{B(x)}$, the domain is defined by the restriction$B(x) eq 0\n- **Irrational Functions (Square Roots)**: For a function f(x) = \sqrt{A(x)}$, the domain is defined by the restriction$A(x) \geq 0\n\n## Domain Applications\n\n### Problem 1\nFunction: f(x) = \frac{2x}{x - 1}\n- Restriction: Denominator cannot be zero: x - 1 eq 0 \implies x eq 1\n- Domain of definition: x \in ]-\infty, 1[ \,\cup\, ]1, +\infty[\n\n### Problem 2\nFunction: f(x) = \frac{x - 1}{x^2 - 3x + 2}\n- Restriction: x^2 - 3x + 2 eq 0\n- Denominator roots: x' = 2$,$x'' = 1\n- Domain of definition: x \in ]-\infty, 1[ \,\cup\, ]1, 2[ \,\cup\, ]2, +\infty[\n\n### Problem 3\nFunction: f(x) = \frac{2x}{x^2 + 1}\n- Restriction: x^2 + 1 eq 0\n- Analysis: Since x^2 \geq 0$for all real numbers,$x^2 + 1 > 0$holds for all$\forall x \in \mathbb{R}.\n- Domain of definition: x \in ]-\infty, +\infty[\n\n\n\n\n# Limits of Functions\n\n## Fundamental Limits at Infinity\n\n- Limit of power functions: \lim_{x \to \pm\infty}(x^2) = +\infty (even degree terms always yield positive infinity)\n- Limit of square root functions: \lim_{x \to \pm\infty}(\sqrt{x}) = \sqrt{+\infty} = +\infty\n- Reciprocal limit: \lim_{x \to \pm\infty}\left(\frac{1}{x}\right) = 0\n\n## Limits of Polynomial Functions at Infinity\n\nFor any polynomial function P(x)$, its limit as$x \to \pm\infty$is determined solely by the limit of its highest-degree term$a x^n:\n\lim_{x \to \pm\infty}(P(x)) = \lim_{x \to \pm\infty}(a x^n)\n\n### Example\nGiven f(x) = 3x^3 + 2x - 1:\n\lim_{x \to -\infty}(f(x)) = \lim_{x \to -\infty}(3x^3) = 3(-\infty)^3 = -\infty\n\n## Limits of Rational Functions at Infinity\n\nFor a rational function f(x) = \frac{A(x)}{B(x)}$, the limit as$x \to \pm\infty$is equal to the limit of the ratio of the highest-degree term of$A(x)$to the highest-degree term of$B(x):\n\lim_{x \to \pm\infty}(f(x)) = \lim_{x \to \pm\infty}\left(\frac{\text{highest degree term of } A(x)}{\text{highest degree term of } B(x)}\right)\n\n## One-Sided Limits at a Specific Point\n\nFor a function f(x)$evaluated at a point$a:\n- **Left-hand limit**: \lim_{x \to a^-}(f(x))$represents the limit to the left of$f$at$a.\n- **Right-hand limit**: \lim_{x \to a^+}(f(x))$represents the limit to the right of$f$at$a.\n\n## Quotient Operations Involving Zero and Infinity\n\nWhen evaluating limits involving non-zero real constants N, zero, and infinity:\n\n### Positive Numerator (N > 0)\n- \frac{N}{0^+} = +\infty\n- \frac{N}{0^-} = -\infty\n\n### Negative Numerator (N < 0)\n- \frac{N}{0^+} = -\infty\n- \frac{N}{0^-} = +\infty\n\n### General Limit Quotients\n- \frac{\text{number}}{\infty} = 0\n- \frac{\text{number}}{0} = \infty\n- \frac{\infty}{\text{number}} = \infty\n- \frac{0}{\text{number}} = 0\n\n\n# Indeterminate Forms and Resolution Techniques\n\n## The Four Indeterminate Forms\n\nThe four indeterminate forms (I.F.) in limit evaluations are:\n1. \frac{0}{0}\n2. \frac{\infty}{\infty}\n3. +\infty - \infty\n4. 0 \times \infty\n\n## Methods for Resolving Indeterminate Forms\n\n### Form \frac{0}{0}\n- Apply L'Hôpital's rule\n- Factorize numerator and denominator to cancel common factors\n- Multiply by the conjugate or perform reciprocal transformations\n\n### Form \frac{\infty}{\infty}\n- Keep only the highest-degree terms in numerator and denominator\n- Apply L'Hôpital's rule\n\n### Form +\infty - \infty\n- Find a common denominator\n- Factorize out dominant terms\n- Multiply by the conjugate or expand expressions\n\n### Form 0 \times \infty\n- Transform into a quotient by taking the reciprocal of one factor (converting to \frac{0}{0}$or$\frac{\infty}{\infty})\n- Expand the expression\n\n## Factorization of Second-Degree Polynomials\n\nTo simplify rational expressions and eliminate indeterminate forms, a second-degree polynomial y = ax^2 + bx + c$with roots$x'$and$x'' can be rewritten in factorized form as:\ny = a(x - x')(x - x'')$$