Find equation: lim x → 0 tan 2 ⁡ x − arctan 2 ⁡ x x 4

Question

Find equation:      lim          x      →      0                          tan        2            ⁡      x      −              arctan        2            ⁡      x              x      4

Bruce Townsend · Accepted Answer

One may use some standard Taylor expansions as

x \to 0

We have

\tan x = x + \frac{x^{3}}{3} + O (x^{5})

giving

\tan^{2} x = x^{2} + \frac{2 x^{4}}{3} + O (x^{6})

and

\arctan x = x - \frac{x^{3}}{3} + O (x^{5})

giving

\arctan^{2} x = x^{2} - \frac{2 x^{4}}{3} + O (x^{6}) .

Then

\frac{\tan^{2} x - \arctan^{2} x}{x^{4}} = \frac{4}{3} + O (x^{2})

and

lim_{x \to 0} \frac{\tan^{2} x - \arctan^{2} x}{x^{4}} = \frac{4}{3} .

Lesly Weiss · Accepted Answer

Note that this limit cannot be solved without the use of Taylor series or L'Hospital's Rule. A solution via Taylor series is already presented by Olivier Oloa and I present one with L'Hospital's Rule.
Note that if one wishes to apply L'Hospital's Rule to solve any limit one should try to apply it only when needed. Thus we first make use of algebraic simplification and standard limits and then use L'Hospital's Rule.
We have

\begin{aligned} L & = lim_{x \to 0} \frac{\tan^{2} x - \arctan^{2} x}{x^{4}} \\ = lim_{x \to 0} \frac{\tan x + \arctan x}{x} \cdot \frac{\tan x - \arctan x}{x^{3}} \\ = lim_{x \to 0} (\frac{\tan x}{x} + \frac{\arctan x}{x}) \cdot \frac{\tan x - \arctan x}{x^{3}} \\ = lim_{x \to 0} (1 + 1) \cdot \frac{\tan x - \arctan x}{x^{3}} \\ = 2 lim_{x \to 0} \frac{\tan x - x + x - \arctan x}{x^{3}} \\ = 2 lim_{x \to 0} \frac{\tan x - x}{x^{3}} + \frac{x - \arctan x}{x^{3}} \\ (1) & = 2 lim_{x \to 0} \frac{\tan x - x}{x^{3}} + 2 lim_{x \to 0} \frac{x - \arctan x}{x^{3}} \\ = 2 lim_{x \to 0} \frac{\tan x - x}{x^{3}} + 2 lim_{x \to 0} \frac{x - \arctan x}{\arctan^{3} x} \cdot \frac{\arctan^{3} x}{x^{3}} \\ = 2 lim_{x \to 0} \frac{\tan x - x}{x^{3}} + 2 lim_{x \to 0} \frac{x - \arctan x}{\arctan^{3} x} \cdot 1 \\ = 2 lim_{x \to 0} \frac{\tan x - x}{x^{3}} + 2 lim_{t \to 0} \frac{\tan t - t}{t^{3}} (putting t = \arctan x) \\ (2) & = 4 lim_{x \to 0} \frac{\tan x - x}{x^{3}} \\ = 4 lim_{x \to 0} \frac{\sec^{2} x - 1}{3 x^{2}} (via L'Hospital's Rule) \\ = \frac{4}{3} lim_{x \to 0} \frac{\tan^{2} x}{x^{2}} \\ = \frac{4}{3} \end{aligned}

Steps from (1) to (2) are based on the assumption that the limit

lim_{x \to 0} \frac{\tan x - x}{x^{3}}

exists and this assumption is justified in the steps after (2) via L'Hospital's Rule.
Note that L'Hospital's Rule has been used only once (compared to what OP mentions in his post) and the standard limit

lim_{x \to 0} \frac{\tan x}{x} = 1

or equivalently

lim_{x \to 0} \frac{\arctan x}{x} = 1

has been used in the above solution.

Find equation: <munder> <mo movablelimits="true" form="prefix">lim <mrow class="MJX-TeXA

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