Lecture 8: Bernoulli/De l'Hospital Rule and Taylor Series

Lecture 8: Bernoulli and de l’Hospital Rule; Taylor Series

Course Information: - Topic: Regel von Bernoulli und de l’Hospital; Taylorreihen (Mathematics I). - Date: June 3, 2025. - Lecturer: Stephan Block – AG Bionanogrenzflächen, Institute for Physical and Theoretical Chemistry, Department of BCP, Freie Universität Berlin.
Learning Objectives: - Mastery of limit calculations for indeterminate forms such as $\text{lim}{x \rightarrow x_0} \frac{0}{0}$ , $\text{lim}{x \rightarrow x_0} \frac{\infty}{\infty}$ , $\text{lim}{x \rightarrow \infty} \frac{0}{0}$ , and $\text{lim}{x \rightarrow \infty} \frac{\infty}{\infty}$ . - Recognition that specific functions can be represented through Taylor polynomials. - Construction of Taylor polynomials using the principles of differential calculus.

Limit Calculation via Bernoulli and de l’Hospital

Motivation Part 1: The Sinc Function: - In spectroscopy and other fields, one frequently encounters expressions where both the numerator and denominator approach zero or infinity simultaneously. - Example: The Sinc function (Spaltfunktion), defined as $\text{si}(x) = \frac{\sin(x)}{x}$ . - The Problem: As $x \rightarrow 0$ , both $\sin(x)$ and $x$ approach $0$ . This raises the question of whether the limit $\text{lim}_{x \rightarrow 0} \frac{\sin(x)}{x}$ exists. - The Answer: The limit does exist, and its value can be easily proven using the Bernoulli and de l’Hospital rule.
Introductory Example (Polynomials): - Before applying the rule to complex functions, consider a simpler rational function: $\text{lim}_{x \rightarrow 2} \frac{5x^2 - 10}{3x^2 - 6}$ . - Direct substitution of $x = 2$ results in $\frac{5(2)^2 - 10}{3(2)^2 - 6} = \frac{20 - 10}{12 - 6} = \frac{10}{6} = \frac{5}{3}$ . - Note: In this specific case, direct evaluation was possible, but it serves as a baseline for understanding ratios of growth.

Formal Definition and Proof of the Rule

Conditions for Application: - Let $f(x)$ and $g(x)$ be two differentiable functions. - Both functions must share a root (zero) at the same point $x_0$ in their domain: $f(x_0) = g(x_0) = 0$ . - The first derivatives, $f'(x_0)$ and $g'(x_0)$ , must have non-zero values at $x_0$ .
The Rule: - The limit of the ratio of the functions is equal to the limit of the ratio of their derivatives: - $\text{lim}{x \rightarrow x_0} \frac{f(x)}{g(x)} = \text{lim}{x \rightarrow x_0} \frac{f'(x)}{g'(x)} = \frac{f'(x_0)}{g'(x_0)}$
Mathematical Proof (Infinitesimal Approach): - Due to differentiability, in the infinitesimal vicinity of the root $x_0$ , the following substitution holds: - $f(x) = f(x_0) + df = f(x_0) + f'(x_0) \times dx$ - $g(x) = g(x_0) + dg = g(x_0) + g'(x_0) \times dx$ - Since $f(x_0) = 0$ and $g(x_0) = 0$ , the ratio simplifies to: - $\frac{f(x)}{g(x)} = \frac{0 + f'(x_0) \times dx}{0 + g'(x_0) \times dx} = \frac{f'(x_0)}{g'(x_0)}$
Practical Examples: - The Sinc Function: $\text{lim}{x \rightarrow 0} \frac{\sin(x)}{x}$ - Numerator derivative: $\frac{d}{dx} \sin(x) = \cos(x)$ - Denominator derivative: $\frac{d}{dx} x = 1$ - Result: $\text{lim}{x \rightarrow 0} \frac{\cos(x)}{1} = \frac{\cos(0)}{1} = \frac{1}{1} = 1$ - Polynomial Ratio: $\text{lim}{x \rightarrow 2} \frac{5x^2 - 10}{3x^2 - 6}$ - Applying the rule: $\text{lim}{x \rightarrow 2} \frac{10x}{6x} = \frac{10}{6} = \frac{5}{3}$

Advanced Remarks on de l’Hospital’s Rule

Iterative Application: - Sometimes the first application of the rule still results in an indeterminate form (e.g., $0/0$ ). In such cases, the rule must be applied repeatedly until the denominator's derivative is non-zero at $x_0$ , or until the conditions for the rule are no longer met.
Applicability to Infinity: - The rule is equally valid if both functions approach infinity ( $\infty$ ) at the point $x_0$ . - It is irrelevant whether $x_0$ itself is a finite value or located at infinity ( $\pm \infty$ ).
Invalid Forms: - If the limit does not take the form $0/0$ or $\infty/\infty$ , the rule cannot be applied. In these cases, the rule provides no information regarding the existence or non-existence of the limit.

Approximation of Functions via Taylor Polynomials

Motivation: Beyond Local Linearization: - Function reconstruction often involves summing small steps ( $df = f'(x) \times dx$ ). While this works for infinitesimal changes, it becomes inaccurate over larger distances. - To improve accuracy, one can incorporate higher-order derivatives (e.g., 2nd derivative for curvature).
Definition: Taylor Polynomial: - Let $f(x)$ be a function that is sufficiently differentiable. The function can be approximated by: - $f(x) = P_n(x) + R_n(x)$ - Where $P_n(x)$ is the Taylor Polynomial of degree $n$ : - $P_n(x) = \sum_{k=0}^{n} \frac{1}{k!} f^{(k)}(x_0) \times (x - x_0)^k$ - Here, $f^{(k)}(x_0)$ denotes the $k$ -th derivative of the function at the expansion point (Aufpunkt) $x_0$ . - $R_n(x)$ is the remainder (Restglied), representing the error of the approximation.
From Polynomial to Series: - As $n \rightarrow \infty$ , if the remainder $R_n(x)$ converges to zero, the polynomial becomes a Taylor Series. - Real Analytic Functions: A function is called real analytic if its remainder is a null sequence ( $\text{lim}{n \rightarrow \infty} R_n(x) = 0$ ), meaning the function is represented exactly by its Taylor Series: - $f(x) = \sum{k=0}^{\infty} \frac{1}{k!} f^{(k)}(x_0) \times (x - x_0)^k$

Derivative Tables and Common Taylor Series Extensions

Derivatives at Expansion Point $x_0 = 0$ : - Function $f(x) = x^2$ : - $f(0) = 0$ - $f'(0) = 0$ - $f''(0) = 2$ - $f^{(n)}(0) = 0$ for n > 2 - Function $f(x) = \exp(x)$ : - All derivatives $f^{(n)}(0) = \exp(0) = 1$ - Function $f(x) = \sin(x)$ : - $f(0) = 0$ , $f'(0) = 1$ , $f''(0) = 0$ , $f'''(0) = -1$ , $f^{(4)}(0) = 0$ - Function $f(x) = \cos(x)$ : - $f(0) = 1$ , $f'(0) = 0$ , $f''(0) = -1$ , $f'''(0) = 0$ , $f^{(4)}(0) = 1$
Established Taylor Series Formulas: - Sine: $\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dots = \sum_{k=0}^{\infty} (-1)^k \frac{x^{2k+1}}{(2k+1)!}$ - Cosine: $\cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \dots = \sum_{k=0}^{\infty} (-1)^k \frac{x^{2k}}{(2k)!}$ - Exponential: $\exp(x) = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots = \sum_{k=0}^{\infty} \frac{x^k}{k!}$ - Logarithm: $\ln(x+1) = (x-1) - \frac{1}{2}(x-1)^2 + \frac{1}{3}(x-1)^3 - \dots = \sum_{k=1}^{\infty} (-1)^{k+1} \frac{(x-1)^k}{k}$
Visual Convergence: - Graphic analysis (e.g., logic provided by Dirk Andrae) shows that the sine function is approximated better as the number of terms in the polynomial increases, which is particularly vital for larger values of $|x|$ .

The Euler Formula Derivation

Expansion with Complex Numbers: - Taylor series rely only on addition and multiplication, permitting the substitution of complex numbers for $x$ . - Substituting $ix$ into the exponential series: - $\exp(ix) = 1 + (ix) + \frac{(ix)^2}{2!} + \frac{(ix)^3}{3!} + \frac{(ix)^4}{4!} + \frac{(ix)^5}{5!} + \dots$ - Utilizing the properties of the imaginary unit ( $i^2 = -1$ , $i^3 = -i$ , $i^4 = 1$ ): - $\exp(ix) = 1 + ix - \frac{x^2}{2!} - \frac{ix^3}{3!} + \frac{x^4}{4!} + \frac{ix^5}{5!} - \dots$
Separation into Sine and Cosine: - Grouping real and imaginary terms: - $\exp(ix) = \left( 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \dots \right) + i \left( x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dots \right)$ - This proves Euler's Formula: $\exp(ix) = \cos(x) + i \sin(x)$ .

Applications and Practical Utility

Numerical Computation: - Taylor series allow for the calculation of complex function values that would otherwise be difficult to determine manually or digitally.
Simplification in Science: - Taylor polynomials are frequently used to simplify physico-chemical equations, often by truncating the series to the first or second order when variables are small (small-angle approximations, etc.).