1.3 Q-3

Question Statement

Evaluate the following limits:

i. $\quad \lim _{h \rightarrow 0} \frac{\sin 7 x}{x}$

ii. $\quad \lim _{x \rightarrow 0} \frac{\sin x^{0}}{x}$

iii. $\quad \lim _{x \rightarrow 0} \frac{1-\cos \theta}{\sin \theta}$

iv. $\quad \lim _{x \rightarrow \pi} \frac{\sin \theta}{\pi-x}$

v. $\quad \lim _{x \rightarrow 0} \frac{\sin a x}{\sin b x}$

vi. $\quad \lim _{x \rightarrow 0} \frac{x}{\tan x}$

vii. $\quad \lim _{x \rightarrow 0} \frac{1-\cos 2 x}{x^{2}}$

viii. $\quad \lim _{x \rightarrow 0} \frac{1-\cos x}{\sin ^{2} x}$

ix. $\quad \lim _{\theta \rightarrow 0} \frac{\sin ^{2} \theta}{\theta}$

x. $\quad \lim _{x \rightarrow 0} \frac{\sec x-\cos x}{x}$

xi. $\quad \lim _{\theta \rightarrow 0} \frac{1-\cos p \theta}{1-\cos q \theta}$

xii. $\quad \lim _{\theta \rightarrow 0} \frac{\tan \theta-\sin \theta}{\sin ^{3} \theta}$

Background and Explanation

The problem involves evaluating several limits, primarily related to trigonometric functions. Some key concepts include:

• Trigonometric Limits: Basic limits of sine and cosine functions as $x \to 0$.
• Limit Laws: Using properties such as the limit of sine and cosine functions and using substitution methods to simplify the expressions.
• Algebraic Manipulations: Simplifying expressions through common techniques like factoring, multiplying by conjugates, or using standard limit properties.

Solution

i. $\lim _{h \rightarrow 0} \frac{\sin 7 x}{x}$

We can rewrite this as:

$$\lim _{h \rightarrow 0} \frac{\sin 7 x}{\frac{7 x}{7}} = 7 \lim _{h \rightarrow 0} \frac{\sin 7 x}{7 x} = 7(1) = 7$$

ii. $\lim _{x \rightarrow 0} \frac{\sin x^{0}}{x}$

Since $x^0 = 1$ for any $x$:

$$= \lim _{x \rightarrow 0} \frac{\sin \frac{x \pi}{180}}{x} \cdot \frac{\frac{\pi}{180}}{\frac{\pi}{180}} = \lim _{x \rightarrow 0} \frac{\sin \frac{x \pi}{180}}{\frac{x \pi}{180}} \cdot \frac{\pi}{180} = 1 \cdot \frac{\pi}{180} = \frac{\pi}{180}$$

iii. $\lim _{x \rightarrow 0} \frac{1-\cos \theta}{\sin \theta}$

We multiply by the conjugate:

$$\lim _{x \rightarrow 0} \frac{1-\cos \theta}{\sin \theta} \cdot \frac{1+\cos \theta}{1+\cos \theta} = \lim _{x \rightarrow 0} \frac{\sin ^2 \theta}{\sin \theta (1+\cos \theta)} = \lim _{x \rightarrow 0} \frac{\sin \theta}{1+\cos \theta} = \frac{0}{2} = 0$$

iv. $\lim _{x \rightarrow \pi} \frac{\sin \theta}{\pi-x}$

Let $x = \pi - \theta$ so that as $x \to \pi$, $\theta \to 0$:

$$= \lim _{\theta \rightarrow 0} \frac{\sin (\pi - \theta)}{\pi - (\pi - \theta)} = \lim _{\theta \rightarrow 0} \frac{\sin \theta}{\theta} = 1$$

v. $\lim _{x \rightarrow 0} \frac{\sin a x}{\sin b x}$

This is simplified as:

$$= \lim _{x \rightarrow 0} \left( \frac{a}{b} \right) = \frac{a}{b}$$

vi. $\lim _{x \rightarrow 0} \frac{x}{\tan x}$

Using the fact that $\lim _{x \rightarrow 0} \frac{\sin x}{x} = 1$:

$$= \lim _{x \rightarrow 0} \frac{\cos x}{\frac{\sin x}{x}} = 1$$

vii. $\lim _{x \rightarrow 0} \frac{1-\cos 2 x}{x^2}$

Using the identity $\cos 2x = 2\cos^2 x - 1$:

$$= \lim _{x \rightarrow 0} \frac{2 \sin^2 x}{x^2} = 2 \cdot \lim _{x \rightarrow 0} \frac{\sin x}{x} = 2(1) = 2$$

viii. $\lim _{x \rightarrow 0} \frac{1-\cos x}{\sin^2 x}$

We multiply by the conjugate:

$$= \lim _{x \rightarrow 0} \frac{\sin^2 x}{\sin^2 x (1+\cos x)} = \frac{1}{2}$$

ix. $\lim _{\theta \rightarrow 0} \frac{\sin^2 \theta}{\theta}$

Simplifying:

$$= \lim _{\theta \rightarrow 0} \frac{\sin^2 \theta}{\theta} \cdot \theta = 0$$

x. $\lim _{x \rightarrow 0} \frac{\sec x - \cos x}{x}$

Simplifying the expression:

$$= \lim _{x \rightarrow 0} \frac{1-\cos^2 x}{x \cos x} = 0$$

xi. $\lim _{\theta \rightarrow 0} \frac{1-\cos p \theta}{1-\cos q \theta}$

Using the small angle approximations:

$$= \frac{p^2}{q^2}$$

xii. $\lim _{\theta \rightarrow 0} \frac{\tan \theta - \sin \theta}{\sin^3 \theta}$

Simplifying:

$$= \frac{1}{2}$$

Key Formulas or Methods Used

• Limit of Sine and Cosine: $\lim_{x \to 0} \frac{\sin x}{x} = 1$
• Conjugate Multiplication: Used to simplify trigonometric expressions.
• Small Angle Approximation: Simplifications for $\sin x \approx x$ and $\cos x \approx 1$ as $x \to 0$.

Summary of Steps

1. For each limit, identify trigonometric identities or simplifications (like $\sin x \approx x$ as $x \to 0$).
2. Simplify the expression using known limits or algebraic manipulation (e.g., conjugate multiplication, factoring).
3. Apply the standard limits of trigonometric functions to solve for the final result.
4. Ensure all intermediate steps are logically connected to avoid skipping key steps.

For better understanding refer to video provided on 03_Ex 1.3