\(\displaystyle\lim_{x \rightarrow 0} \frac{x}{\sqrt{1−\cos 4x}}\) किसके बराबर है ?

Question

Question

This question was previously asked in

NDA 02/2022 Mathematics Official Paper (Held On 04 Sep 2022)

Answer 1

प्रयुक्त सूत्र:

1 - cos 2x = 2sin²x

गणना:

दिया गया है, \(\displaystyle\lim_{x \rightarrow 0} \frac{x}{\sqrt{1−\cos 4x}}\)

limx→0 x / √(1 − cos 4x)

⇒ 1 − cos 4x = 2 sin²(2x)

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

⇒ limx→0 x / (√2 × sin(2x))

⇒ 1/√2 × limx→0 x / sin(2x)

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

⇒ 1/√2 × 1/2 × 1 = 1 / (2√2)