Evaluation of Trigonometric Limits
\(\underset{\theta \to 0}{\mathop{Lim}}\,\frac{\sin \theta }{\theta }=1\) (where θ is in radius)
Proof: Consider a circle of radius r. let O be the center of the circle such that \(\angle AOB=\theta \), where θ is measured in radians and its value is very small. Suppose the tangent at A meets OB produced at P. From fig we have

Area of ΔOAB < Area of sector OAB < Area of Δ OAP
\(\frac{1}{2}OA\times OB\sin \theta <\frac{1}{2}{{\left( OA \right)}^{2}}\theta <\frac{1}{2}OA\times OP\),
\(\frac{1}{2}{{r}^{2}}\sin \theta <\frac{1}{2}{{r}^{2}}\theta <\frac{1}{2}{{r}^{2}}\tan \theta \),
In triangle
AP = OA Tanθ
Sinθ < θ < tanθ
sinθ < θ < sinθ/cosθ
\(1<\frac{\theta }{\sin \theta }<\frac{1}{\cos \theta }\),
(since θ is small, sin θ > 0)
\(1>\underset{\theta \to 0}{\mathop{\lim }}\,\frac{\sin \theta }{\theta }>\underset{\theta \to 0}{\mathop{\lim }}\,\cos \theta \),
\(\underset{\theta \to 0}{\mathop{\lim }}\,\cos \theta <\underset{\theta \to 0}{\mathop{\lim }}\,\frac{\sin \theta }{\theta }<1\),
\(\cos 0<\underset{\theta \to 0}{\mathop{\lim }}\,\frac{\sin \theta }{\theta }<1\),
\(1<\underset{\theta \to 0}{\mathop{\lim }}\,\frac{\sin \theta }{\theta }<1\),
By sandwich theorem
\(\underset{\theta \to 0}{\mathop{\lim }}\,\frac{\sin \theta }{\theta }=1\),
Example: Evaluate the \(\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin 3x}{x}\),
Solution:
We have \(\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin 3x}{x}\),
\(\underset{x\to 0}{\mathop{\lim }}\,\frac{3\sin 3x}{3x}\),
\(3\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin 3x}{3x}\),
\(\left( \because \underset{x\to 0}{\mathop{\lim }}\,\frac{\sin 3x}{3x}=1 \right)\),
3(1) = 3