# Evaluation of Trigonometric Limits

## 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