Hi, Theia! Thankyou for a nice solution and your participation!(Yes)Theia said:Something like this(?):
Assume \(\displaystyle |x| \approx 0\). Then
\(\displaystyle \begin{align*} \sin (\arctan x)-\tan (\arcsin x) &= \frac{x}{\sqrt{1 + x^2}} - \frac{x}{\sqrt{1-x^2}} \\
&= x\left( 1 - \frac{x^2}{2} + \cdots \right) - x\left( 1 + \frac{x^2}{2} \right) \\
&= -x^3 + \cdots\end{align*}\)
and
\(\displaystyle \begin{align*} \arcsin (\tan x) - \arctan (\sin x) &= \tan x + \frac{\tan^3 x}{6} + \cdots - \sin x + \frac{\sin^3 x}{3} + \cdots \\
&= x + \frac{x^3}{3} + \cdots +\frac{x^3}{6} + \cdots - \left( x - \frac{x^3}{6} + \cdots \right) + \frac{x^3}{3} + \cdots \\
&= x^3 + \cdots\end{align*}\).
Hence
\[\lim_{x\rightarrow 0}\frac{\sin (\arctan x)-\tan (\arcsin x)}{\arcsin (\tan x)-\arctan (\sin x)} = -1.\]
A limit in trigonometry refers to the value that a trigonometric expression approaches as the input variable gets closer and closer to a specific value. It is denoted as lim x→a f(x), where a is the value that x is approaching and f(x) is the trigonometric expression.
To evaluate the limit of a trigonometric expression, you can use a variety of methods such as substitution, graphing, or trigonometric identities. It is important to simplify the expression as much as possible before evaluating the limit.
Yes, the limit of a trigonometric expression can be undefined if the expression has a vertical asymptote or if the limit does not exist due to oscillation or divergence.
Some common trigonometric limits include lim x→0 sin(x)/x = 1, lim x→0 tan(x)/x = 1, and lim x→π/2 cos(x) = 0. These limits are useful in solving more complex trigonometric limits.
Trigonometric limits can be used to model and solve real-world problems involving angles and periodic functions. For example, they can be used in physics to calculate the velocity of an object in circular motion or in astronomy to determine the position of celestial objects.