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NTA Abhyas 2020
Exam
NTA Abhyas
Year
2020
Subject
Mathematics
Chapter
Limits

NTA Abhyas 2020 · Mathematics · Limits

The value of _ x → 0 ( ( 5x + 5x))/( 3x) is equal to

The value of lim⁡x→0log⁡(sin⁡5x+cos⁡5x)tan⁡3x\lim_{x \to 0}\dfrac{\log\left(\sin 5x + \cos 5x\right)}{\tan 3x} is equal to

Options

  1. A103\dfrac{10}{3}
  2. B203\dfrac{20}{3}
  3. C56\dfrac{5}{6}
  4. D53\dfrac{5}{3}

Correct answer

D. 53\dfrac{5}{3}

Step-by-step solution

lim⁡x→012log⁡(1+sin⁡10x)tan⁡3x(as,1+sin⁡10x=(sin⁡5x+cos⁡5x)2)\lim_{x \to 0}\dfrac{1}{2}\dfrac{\log\left(1 + \sin 10x\right)}{\tan 3x} ( as, 1 + \sin 10x = ( \sin 5x + \cos 5x{)}^{2} )
=lim⁡x→012log⁡(1+sin⁡10x)(sin⁡10x)⋅(sin⁡10x10x)⋅(3xtan⁡3x)⋅10x3x= \lim_{x \to 0}\dfrac{1}{2}\log ( 1 + \sin 10x )\left(\sin 10x\right) ⋅ \left(\dfrac{\sin 10x}{10x}\right) ⋅ \left(\dfrac{3x}{\tan 3x}\right) ⋅ \dfrac{10x}{3x}
=12×1×1×103=53= \dfrac{1}{2} \times 1 \times 1 \times \dfrac{10}{3} = \dfrac{5}{3}
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