Evaluation of Trigonometric Limits

Evaluation of Trigonometric Limits

$$\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin \theta }{\theta }=1$$ (where θ is in radians).

Proof:

1. Consider a circle of radius such that ∠AOB = θ, where θ is measured in radius and its value is very small. Suppose the tangent at A meets. OB produced at P. from Fig. we have

Area of ΔABC < area of sector OAB < Area of ΔOAP

½ x OA x OB sinθ < ½ (OA)² θ < ½ OA x AP.

½ x r² sinθ < ½ r²θ < ½ r² tanθ

Sinθ < θ < tanθ

1 < θ/sinθ < 1/cosθ

1 > sinθ/θ > cosθ

i.e., $$1>\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin \theta }{\theta }>\underset{x\to 0}{\mathop{\lim }}\,\cos \theta$$.

(or)

$$\underset{x\to 0}{\mathop{\lim }}\,\cos \theta <\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin \theta }{\theta }<1$$.

$$1<\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin \theta }{\theta }<1$$.

By sandwich theorem

$$\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin \theta }{\theta }=1$$.

2. $$\underset{x\to 0}{\mathop{\lim }}\,\frac{\tan \theta }{\theta }=1$$.

We have $$\underset{x\to 0}{\mathop{\lim }}\,\frac{\tan \theta }{\theta }$$.

$$\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin \theta /\cos \theta }{\theta }$$.

$$\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin \theta }{\theta }.\underset{x\to 0}{\mathop{\lim }}\,\cos \theta$$.

Since $$\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin \theta }{\theta }=1$$.

= (1) (1) = 1

3. $$\underset{x\to a}{\mathop{\lim }}\,\frac{\sin (\theta -a)}{(\theta -a)}=1$$.

We have

$$\underset{h\to 0}{\mathop{\lim }}\,\frac{\sin (a+h-a)}{(a+h-a)}$$.

$$\underset{h\to 0}{\mathop{\lim }}\,\frac{\sin (h)}{h}$$.

Since $$\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin \theta }{\theta }=1$$.

= 1

4. $$\underset{x\to a}{\mathop{\lim }}\,\frac{\tan (\theta -a)}{(\theta -a)}=1$$.

5. $$\underset{x\to 0}{\mathop{\lim }}\,\frac{{{\sin }^{-1}}x}{x}=1$$.

6. $$\underset{x\to 0}{\mathop{\lim }}\,\frac{{{\operatorname{Tan}}^{-1}}x}{x}=1$$.

Example: Evaluate the following $$\underset{x\to 0}{\mathop{\lim }}\,\frac{\operatorname{Sin}3x}{x}$$.

Solution: Given that $$\underset{x\to 0}{\mathop{\lim }}\,\frac{\operatorname{Sin}3x}{x}$$.

$$\underset{x\to 0}{\mathop{\lim }}\,\frac{\operatorname{Sin}3x}{x}\times \frac{3}{3}$$.

$$\underset{x\to 0}{\mathop{\lim }}\,3.\frac{\operatorname{Sin}3x}{3x}$$.

$$3.\underset{x\to 0}{\mathop{\lim }}\,\frac{\operatorname{Sin}3x}{3x}$$.

= 3 (1) = 3