What is \(\displaystyle\lim_{x \rightarrow 0} \frac{x}{\sqrt{1−\cos 4x}}\) equal to ?

Concept:

Limit of Trigonometric Functions:

• To evaluate limits involving trigonometric functions near zero, we use standard limits and trigonometric identities.
• Standard Limit: lim_x→0 (sin x / x) = 1
• Trigonometric Identity: 1 − cos θ = 2 sin²(θ/2)
• These help simplify expressions that are otherwise indeterminate forms like 0/0.
• Use rationalization or substitution to reduce complex expressions into standard limits.

Calculation:

Given,

Expression: lim_x→0 x / √(1 − cos 4x)

⇒ 1 − cos 4x = 2 sin²(2x)

⇒ √(1 − cos 4x) = √(2 sin²(2x)) = √2 × sin(2x)

⇒ lim_x→0 x / (√2 × sin(2x))

⇒ 1/√2 × lim_x→0 x / sin(2x)

⇒ 1/√2 × lim_x→0 1 / (2 × (sin(2x)/2x))

⇒ 1/√2 × 1/2 × 1 = 1 / (2√2)

∴ The value of the limit is 1 / (2√2).