Question

The figure shows a circular arc of length s and a chord of length d , both subtended by a central angle\(\theta \) . Find

\({\lim _{\theta \to {0^ + }}}\frac{s}{d}\)

Short answer

The limit of the above function is 1.

Step-by-step solution

Precise Definition of differentiation

A derivative is used in calculus mathematics.The arc of the circle is connecting between distinct points, this is called circular arc also. An angle at the center of the circle is less than \(\pi \) radian is minor arc and another angle is greater than \(\pi \) radian is called a major arc.

Simplify the obtained condition

Let \(s = r\theta \) and \(d = 2r\sin \frac{\theta }{2}\).

Put the value of s and d in the limit equation.

\begin{aligned}{\lim _{\theta \to 0}}\frac{{r\theta }}{{2r\sin {\raise0.7ex\hbox{$\theta $} \!\mathord{\left/{\vphantom {\theta 2}}\right.}\!\lower0.7ex\hbox{$2$}}}} &= \frac{1}{2}{\lim _{\theta \to 0}}\frac{\theta }{{\sin {\raise0.7ex\hbox{$\theta $} \!\mathord{\left/{\vphantom {\theta 2}}\right.}\!\lower0.7ex\hbox{$2$}}}} \\ &= {\lim _{\theta \to 0}}\frac{{{\raise0.7ex\hbox{$\theta $}\!\mathord{\left/{\vphantom {\theta 2}}\right.}\!\lower0.7ex\hbox{$2$}}}}{{\sin {\raise0.7ex\hbox{$\theta $} \!\mathord{\left/{\vphantom {\theta 2}}\right.}\!\lower0.7ex\hbox{$2$}}}} \\ & = 1 \\ \end{aligned}