Video Notes on Limits and Continuity

Function Descriptions

• $f_1(x) = \sqrt{x + 13}$
  • This function, $f_1(x)$, calculates the square root of the value $x$ plus 13.
  • Domain: $x \ge -13$
  • Graph notes: starts at the point (-13, 0) and increases as x grows; for example, $f_1(3) = \sqrt{3 + 13} = 4$
  • Shape: increasing, concave down (typical of a square-root function)
• $f_2(x) = \dfrac{1}{x + 2}$
  • This function, $f_2(x)$, is a rational function, meaning it's a ratio of two polynomials. It calculates 1 divided by the sum of $x$ and 2.
  • Graph notes: hyperbola with a vertical asymptote at $x = -2$; on the right side ($x > -2$) the branch is positive and descends toward 0 as $x \to \infty$; on the left side ($x < -2$) the branch is negative and ascends toward 0 as $x \to -\infty$

Limits at x = -2 for the rational function

• Consider $f_2(x) = \dfrac{1}{x + 2}$
• Right-hand limit: $\lim_{x \to -2^+} \dfrac{1}{x + 2} = +\infty$
• Left-hand limit: $\lim_{x \to -2^-} \dfrac{1}{x + 2} = -\infty$
• General limit: $\lim_{x \to -2} \dfrac{1}{x + 2}$ does not exist (infinite discontinuity)
• Interpretation: vertical asymptote at $x = -2$; the function values blow up to opposite infinities on each side

Holes, holes filled, and continuity nuances

• The transcript mentions a point where an open circle would be, but the point is filled in by another value
• Conceptual meaning: this is related to removable discontinuities
• If a graph would have a hole at $x = a$ (limit exists) but the function value $f(a)$ is defined differently or not defined, the function is not continuous at $a$
• If a filled dot at $x = a$ matches the limit value, the function is continuous at $a$
• If the filled dot does not match the limit, there is a removable discontinuity or a jump depending on the configuration

Continuity of elementary functions

• Sine and cosine are continuous everywhere
  • Intuition from the transcript: you can draw sin and cos without lifting your pencil; that matches the formal idea that they are continuous for all real $x$
  • Formal statement: for all $x \in \mathbb{R}$, $\lim<em>{h \to 0} \sin(x + h) = \sin x$ and $\lim</em>{h \to 0} \cos(x + h) = \cos x$
  • Therefore, sin $x$ and cos $x$ are continuous on $\mathbb{R}$
• Polynomials are continuous everywhere
  • Quote from the transcript: "polynomials were continuous everywhere"
  • Formalization: if $p(x)$ is a polynomial, then for every $a$, $\lim_{x \to a} p(x) = p(a)$. This means that for any polynomial representation $p(x)$, the limit of the function as $x$ approaches a point $a$ is simply the function's value at that point $a$.
• Rational functions and domain constraints
  • A rational function $r(x) = \dfrac{p(x)}{q(x)}$ (a fraction where $p(x)$ and $q(x)$ are polynomials, and $q(x)$ is not zero) is continuous at a point $a$ provided $q(a) \neq 0$
  • If $q(a) = 0$, the function is not continuous at $a$ (the domain excludes $a$)
  • The transcript notes: "this is a rational function. Clearly, it's not continuous everywhere. Negative two fails. This is not continuous everywhere because there's anything exist."—emphasizing the vertical asymptote at $x = -2$ where the denominator is zero

End behavior and infinite limits

• Infinite behavior near vertical asymptotes
  • When a function tends to $\pm \infty$ near a point, that indicates a vertical asymptote and an improper limit
  • Example from the transcript: near $x = -2$ for $f_2$, the function heads toward $\text{-}\infty$ from the left and $+\infty$ from the right
  • Consequence: the overall limit at that point does not exist

Takeaways and connections to foundational ideas

• To determine limit existence at a point, compare one-sided limits:
  • If both sides approach the same finite value $L$, then $\lim_{x \to a} f(x) = L$ and the function is continuous at $a$ (provided $f(a) = L$)
  • If the one-sided limits diverge to opposite infinities or do not agree, the limit does not exist
• Core definitions touched on in the transcript
  • Continuity: a function is continuous at a point $a$ if $\lim_{x \to a} f(x) = f(a)$
  • Vertical asymptotes: points where the function diverges to $\pm \infty$ as $x$ approaches a given value
  • Holes and removable discontinuities: a hole indicates a point not in the domain; a filled dot at the same $x$-value may or may not match the limit value

Formulas and notation used

• Square-root function: $f_1(x) = \sqrt{x + 13}$ This function means the square root of $x$ plus 13. Its domain is $x \ge -13$, so $x$ must be greater than or equal to -13.
• Example value: $f_1(3) = \sqrt{3 + 13} = 4$ This is an example of the square root of $3$ plus $13$, which equals $4$.
• Rational function: $f_2(x) = \dfrac{1}{x + 2}$ This function means $1$ divided by the quantity $x$ plus $2$.
• Vertical asymptote at: $x = -2$ This marks the vertical asymptote where $x$ equals -2.
• One-sided limits for the rational function: $\lim<em>{x \to -2^-} \dfrac{1}{x + 2} = -\infty$ (The left-hand limit as $x$ approaches -2 from values less than -2, results in negative infinity), $\lim</em>{x \to -2^+} \dfrac{1}{x + 2} = +\infty$ (The right-hand limit as $x$ approaches -2 from values greater than -2, results in positive infinity).
• Limit at the vertical asymptote does not exist: $\lim_{x \to -2} \dfrac{1}{x + 2} \text{ does not exist}$ (The general limit as $x$ approaches -2 does not exist because the one-sided limits approach different infinities).
• Sine and cosine continuity statements: $\text{For all } x \in \mathbb{R}, \quad \lim<em>{h \to 0} \sin(x + h) = \sin x, \quad \lim</em>{h \to 0} \cos(x + h) = \cos x$ (These statements show that for all real numbers $x$, the limit of $\sin(x + h)$ as $h$ approaches zero is $\sin x$, and similarly for $\cos(x + h)$ is $\cos x$, demonstrating their continuity).
• Polynomials: continuous on $\mathbb{R}$ (all real numbers); rational functions continuous where the denominator is nonzero.

Connections to lectures, real-world relevance, and implications

• Continuity is a foundational concept in calculus, underpinning derivatives and integrals
• Understanding where functions are continuous helps in graphing, evaluating limits, and solving problems involving composition of functions
• The idea of holes vs filled points connects to the concept of function definition on a domain and how the limit behavior relates to the actual function value at a point