NTA Abhyas 2020
Exam NTA Abhyas
Year 2020
Subject Mathematics
Chapter Limits NTA Abhyas 2020 · Mathematics · Limits
The value of _ x → 0 1 x ^ 18 (1 - cos ( x ^ 3 3 ) - ( x ^ 6 6 ) +...
The value of
lim x → 0 1 x 18 \lim_{x \to 0} \dfrac{1}{{x}^{18}} lim x → 0 x 18 1 \left(1 - cos
\left(\dfrac{{x}^{3}}{3}\right) - \cos \left(\dfrac{{x}^{6}}{6}\right) + \cos \left(\dfrac{{x}^{3}}{3}\right) ⋅ \cos \left(\dfrac{{x}^{6}}{6}\right)\right) is
λ 2 {\lambda}^{2} λ 2 , then the value of
900 λ 900\lambda 900 λ is equal to (here,
λ > 0 \lambda > 0 λ > 0 )
Step-by-step solution
lim x → 0 { 1 − cos ( x 3 3 ) − cos ( x 6 6 ) ( 1 − cos ( x 3 3 ) ) } ⋅ 1 x 18 \lim_{x \to 0} \left\{1 - \cos \left(\dfrac{{x}^{3}}{3}\right) - \cos \left(\dfrac{{x}^{6}}{6}\right)\left(1 - \cos \left(\dfrac{{x}^{3}}{3}\right)\right)\right\} ⋅ \dfrac{1}{{x}^{18}} lim x → 0 { 1 − cos ( 3 x 3 ) − cos ( 6 x 6 ) ( 1 − cos ( 3 x 3 ) ) } ⋅ x 18 1 lim x → 0 ( 1 − cos ( x 3 3 ) ) ( 1 − cos ( x 6 6 ) ) x 18 \lim_{x \to 0} \dfrac{\left(1 - \cos \left(\dfrac{{x}^{3}}{3}\right)\right)\left(1 - \cos \left(\dfrac{{x}^{6}}{6}\right)\right)}{{x}^{18}} lim x → 0 x 18 ( 1 − cos ( 3 x 3 ) ) ( 1 − cos ( 6 x 6 ) ) lim x → 0 1 − cos ( x 3 3 ) ( x 3 3 ) 2 ( x 6 9 ) ⋅ 1 − cos ( x 6 6 ) ( x 6 6 ) 2 ⋅ ( x 6 6 ) 2 ⋅ 1 x 18 \lim_{x \to 0} \dfrac{1 - \cos \left(\dfrac{{x}^{3}}{3}\right)}{{\left(\dfrac{{x}^{3}}{3}\right)}^{2}}\left(\dfrac{{x}^{6}}{9}\right) ⋅ \dfrac{1 - \cos \left(\dfrac{{x}^{6}}{6}\right)}{{\left(\dfrac{{x}^{6}}{6}\right)}^{2}} ⋅ {\left(\dfrac{{x}^{6}}{6}\right)}^{2} ⋅ \dfrac{1}{{x}^{18}} lim x → 0 ( 3 x 3 ) 2 1 − cos ( 3 x 3 ) ( 9 x 6 ) ⋅ ( 6 x 6 ) 2 1 − cos ( 6 x 6 ) ⋅ ( 6 x 6 ) 2 ⋅ x 18 1 lim x → 0 1 2 × x 6 9 ⋅ 1 2 ⋅ x 12 36 ⋅ 1 x 18 = 1 ( 36 ) 2 \lim_{x \to 0} \dfrac{1}{2} \times \dfrac{{x}^{6}}{9} ⋅ \dfrac{1}{2} ⋅ \dfrac{{x}^{12}}{36} ⋅ \dfrac{1}{{x}^{18}} = \dfrac{1}{{\left(36\right)}^{2}} lim x → 0 2 1 × 9 x 6 ⋅ 2 1 ⋅ 36 x 12 ⋅ x 18 1 = ( 36 ) 2 1 ∴ λ = 1 36 ∴ \lambda = \dfrac{1}{36} ∴ λ = 36 1
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