Magnitude of Quantities and De L'Hospital's Rule

A. Magnitude of Quantities.

The following quantities are arranged in the order of their rate of rise for large x \[ 1 < \ln x < x^\epsilon < x < x^n < e^x \] [Here \(\epsilon\) is a positive number less than 1.]

To ascertain their behavior for small y make the substitution \(y=\dfrac{1}{x}\).

The following approximations hold for small x. \[ \begin{aligned} \sin x &\sim x \\ \cos x &\sim 1 - \dfrac{x^2}{2} \\ \tan^{-1}x &\sim x \sim \tan x \\ \sinh x &\sim x \\ e^x &\sim 1+x \\ (1+x)^n &\sim 1+nx \\ (1+x)^{1/2} &\sim 1-\dfrac{1}{2}x \end{aligned} \]

B. De L’Hospital’s Rule.

If as \(x\to a\), \(f(x)\) and \(g(x) \to 0\) \[ \lim_{x\to a} \dfrac{f(x)}{g(x)} = \lim_{x\to a} \dfrac{f'(x)}{g'(x)} \]

 

Example: \(\lim_{x\to 0} \dfrac{\tan x}{[(1+x)^3-1]}\) \[ = \dfrac{\sec^2 x}{3(1+x)^2} = \dfrac{1}{3} \]

Problem: Find \(\lim_{x\to 0}\)

  1. \(\dfrac{\sin x}{(e^{3x}-1)}\)
  2. \(\dfrac{\ln(\frac{1}{2}+1/2\sqrt{1+x^2})}{(e^{x^2}-\cos x^2)}\)

Problem: Give an approximate expression for \((\sin kr/r^2)-(k \cos kr/r)\) for small r.