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 $1 < \ln x < x^{ϵ} < x < x^{n} < e^{x}$ [Here $ϵ$ is a positive number less than 1.]

To ascertain their behavior for small y make the substitution $y = \frac{1}{x}$ .

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

Theorem 1

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 1.

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

Exercise 1.

Problem: Find $lim_{x \to 0}$

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

Exercise 2.

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

A. Magnitude of Quantities

B. De L’Hospital’s Rule

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