Limits and continuity Knowledge-Based Study

Study notes

1. What is a Limit?

   The limit of a function describes the behavior of the output values as the input values approach a specific real number. We write this as:

   $\lim_{x \to c} f(x) = L$

   This statement means that as $x$ gets arbitrarily close to $c$ from both sides (without ever necessarily equaling $c$), the values of $f(x)$ get arbitrarily close to the real number $L$.

   One-Sided Limits

   A limit exists at $x = c$ if and only if the function approaches the same target value from both directions:

   • Left-hand limit: $\lim_{x \to c^-} f(x)$ represents values of $x$ approaching $c$ where $x < c$.
   • Right-hand limit: $\lim_{x \to c^+} f(x)$ represents values of $x$ approaching $c$ where $x > c$.

   Fundamental Existence Criterion:

   $\lim_{x \to c} f(x) = L$ if and only if $\lim_{x \to c^-} f(x) = L$ and $\lim_{x \to c^+} f(x) = L$.

   Direct Substitution and Limit Laws

   For elementary functions such as polynomials, rational functions, and trigonometric functions within their domains, limits can often be evaluated directly:

   $\lim_{x \to c} f(x) = f(c)$

   Indeterminate Form $\frac{0}{0}$

   When direct substitution produces the indeterminate form $\frac{0}{0}$, algebraic simplification is required before re-evaluating the limit:

   • Factoring: Factor polynomials in the numerator and denominator to cancel common factors that cause division by zero.
   • Rationalization: Multiply the numerator and denominator by the conjugate of a radical expression.
   • Trigonometric Identities: Apply fundamental trigonometric limits such as $\lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1$.

   Example:

   $\lim_{x \to 2} \frac{x^2 - 4}{x - 2} = \lim_{x \to 2} \frac{(x - 2)(x + 2)}{x - 2} = \lim_{x \to 2} (x + 2) = 4$

   Three Conditions for Continuity

   A function $f(x)$ is said to be continuous at a point $x = c$ if the graph passes through $c$ without any interruption, jump, or hole. Formally, all three of the following conditions must hold:

   1. $f(c)$ is defined (meaning $c$ is in the domain of $f$).
   2. $\lim_{x \to c} f(x)$ exists (the left-hand limit equals the right-hand limit).
   3. $\lim_{x \to c} f(x) = f(c)$ (the limiting value equals the actual function value).

   If any of these three conditions is violated, the function is discontinuous at $x = c$.

   A function is described as continuous on an open interval $(a, b)$ if it is continuous at every point in that interval.

   Classifying Discontinuities

   Discontinuities are classified according to how the continuity conditions fail:

   • Removable Discontinuity: The limit $\lim_{x \to c} f(x)$ exists, but $f(c)$ is either undefined or does not equal the limit. This manifests as a hole in the graph.
   • Jump Discontinuity: The one-sided limits $\lim_{x \to c^-} f(x)$ and $\lim_{x \to c^+} f(x)$ both exist as finite numbers, but they are not equal.
   • Essential (Infinite) Discontinuity: At least one of the one-sided limits does not exist as a finite number, frequently approaching $\infty$ or $-\infty$, which produces a vertical asymptote.

   Intermediate Value Theorem (IVT)

   If $f$ is continuous on the closed interval $[a, b]$, and $k$ is any real number strictly between $f(a)$ and $f(b)$, then there exists at least one number $c \in (a, b)$ such that $f(c) = k$.

   This theorem provides a theoretical foundation for proving the existence of roots and solutions to equations.