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L'Hôpital's Rule.

An expert is a person who has made all the mistakes that can be made in a very narrow field, Neils Bohr.

 

Motivation

$\lim_{x \to 1} \frac{x^3-1}{x^2-1} =$[Indeterminate form, 0/0] = $\lim_{x \to 1} \frac{\frac{x^3-1}{x-1}}{\frac{x^2-1}{x-1}}$ [Notice that the numerator is just $\frac{x^3-1}{x-1} = \frac{f(x)-f(1)}{(x-1)}$ →x→1 f’(1) where f(x) = x3-1, f(1) = 0] $\frac{3x^2}{2x} = \frac{3}{2}.$

More generally, let’s assume that f(a) = g(a) = 0, $\lim_{ x \to a}\frac{f(x)}{g(x)} = \lim_{ x \to a}\frac{f(x)/(x-a)}{g(x)/(x-a)}$ =[By assumption, f(a) = g(a) = 0] $\frac{\lim_{ x \to a}\frac{f(x)-f(a)}{(x-a)}}{\lim_{ x \to a}\frac{g(x)-g(a)}{(x-a)}} = \frac{f’(a)}{g’(a)},$ and that works provided that g’(a) ≠ 0.

L’Hôpital’s Rule is a mathematical technique used to evaluate limits of indeterminate forms using derivatives. It states that for functions f and g which are differentiable on an interval I except possible at a point “a” contained in I, if $\lim_{x \to a} f(x) = \lim_{x \to a} g(x) = 0$ or ± ∞, and g’(x) ≠ 0 ∀x ∈ I, except possible at “a”, then $\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f’(x)}{g’(x)},$ provided that the right hand limit exists or equals ± ∞.

$\lim_{x \to 0}\frac{sin(3x)}{sin(2x)}$ =[L’Hôpital’s Rule] $\lim_{x \to 0}\frac{3cos(3x)}{2cos(2x)} =\frac{3cos(0)}{2cos(0)}=\frac{3}{2}.$

When dealing with limits and facing difficulties in directly evaluating them, approximation methods can be deployed to estimate the limit, e.g., sin(u) ≈ u ↭ u ≈ 0 ⇒ $\frac{3cos(3x)}{2cos(2x)}≈\frac{3x}{2x}=\frac{3}{2}.$ Analogously, $\frac{cos(x)-1}{x^2} ≈_{x≈0}\frac{(1-\frac{x^2}{2})-1}{x^2}=\frac{-x^2/2}{x^2}=\frac{-1}{2}.$

$\lim_{x \to 0}\frac{cos(x)-1}{x^2}$ =[L’Hôpital’s Rule] $\lim_{x \to 0}\frac{-sin(x)}{2x}$ = [There may be instances where we would need to apply L’Hôpital’s Rule multiple times] $\lim_{x \to 0}\frac{-cos(x)}{2} = \frac{-1}{2}.$

$\lim_{x \to 0^+} xln(x)$ =[It seems that L’Hôpital’s Rule does not apply, 0·(-∞)] $\lim_{x \to 0^+} \frac{ln(x)}{1/x}$ = [L’Hôpital’s Rule] $\lim_{x \to 0^+} \frac{1/x}{-1/x^2} = \lim_{x \to 0^+} -x = 0$

$\lim_{x \to ∞} xe^{-px} = \lim_{x \to ∞} \frac{x}{e^{px}}$ =[L’Hôpital’s Rule, ∞/∞] $\lim_{x \to ∞} \frac{1}{pe^{px}} = \frac{1}{∞} = 0.$ In words, x grows slower than epx as x → ∞.

Let a > 1, $\lim_{x \to ∞} \frac{e^{px}}{x^a} = (\lim_{x \to ∞} \frac{e^{px/a}}{x})^a$ =[L’Hôpital’s Rule] $(\lim_{x \to ∞} \frac{p}{a}\frac{e^{px/a}}{1})^a = (\frac{∞}{1})^a = ∞.$ In words, the exponential epx grows faster than any power of x as x → ∞.

$\lim_{x \to ∞} \frac{ln(x)}{x^{1/3}}$ =[L’Hôpital’s Rule, ∞/∞] $\lim_{x \to ∞} \frac{1/x}{1/3·x^{-2/3}} =\lim_{x \to ∞} 3x^{2/3-1} =\lim_{x \to ∞} 3x^{-1/3} = 0.$ In words, ln(x) grows slower than $x^{1/3}$ or any arbitrary positive power of x.

$\lim_{x \to 0+} x^x$ =[Recall that ab=ebln(a)] $\lim_{x \to 0+} e^{xln(x)}$. Notice that $\lim_{x \to 0+} xln(x) = \lim_{x \to 0+} \frac{ln(x)}{1/x} = \lim_{x \to 0+} \frac{1/x}{-1/x^2} = \lim_{x \to 0+} -x = 0 ⇒ \lim_{x \to 0+} e^{xln(x)} = e^0 = 1.$

$\lim_{x \to 0+} \frac{sin(x)}{x^2} = \lim_{x \to 0+} \frac{cos(x)}{2x} = \lim_{x \to 0+} \frac{-sin(x)}{2} = 0$. However, using the approximation method sin(x) ≈ x, hence sin(x)/x2 ≈ 1/x → ∞. There is something very fishy🐟🐬, what is it?

This is false because cos(x)/2x indeterminate is of the form “1/0”, and therefore, we are applying L’Hôpital’s Rule wrong, $\lim_{x \to 0+} \frac{cos(x)}{2x} ≠ \lim_{x \to 0+} \frac{-sin(x)}{2}$.

Rates of growth and decay

In mathematics and computer science, the concept of rates of growth often refers to the asymptotic behavior of functions as their input values become (very) large and we express it as f(x) «x→∞ g(x). It means $\lim_{x \to ∞}\frac{f(x)}{g(x)} = 0$, e.g., ln(x) « xp « ex « $e^{x^2}$ as x → ∞, p > 0.

Analogously, 1/ln(x) » 1/xp » e-x » $e^{x^2}$ as x → ∞, p > 0.

Bibliography

This content is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License and is based on MIT OpenCourseWare [18.01 Single Variable Calculus, Fall 2007].
  1. NPTEL-NOC IITM, Introduction to Galois Theory.
  2. Algebra, Second Edition, by Michael Artin.
  3. LibreTexts, Calculus. Abstract and Geometric Algebra, Abstract Algebra: Theory and Applications (Judson).
  4. Field and Galois Theory, by Patrick Morandi. Springer.
  5. Michael Penn, and MathMajor.
  6. Contemporary Abstract Algebra, Joseph, A. Gallian.
  7. YouTube’s Andrew Misseldine: Calculus. College Algebra and Abstract Algebra.
  8. MIT OpenCourseWare 18.01 Single Variable Calculus, Fall 2007 and 18.02 Multivariable Calculus, Fall 2007.
  9. Calculus Early Transcendentals: Differential & Multi-Variable Calculus for Social Sciences.
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