NTA Abhyas 2020
Exam NTA Abhyas
Year 2020
Subject Mathematics
Chapter Limits NTA Abhyas 2020 · Mathematics · Limits
The value of _ x → 0 e^ - (x2)/(2) - cos x x^ 3 tan x is equal to
The value of
lim x → 0 e − x 2 2 − c o s x x 3 t a n x \lim_{x \to 0}\dfrac{e^{- \dfrac{x2}{2}} - cos x}{x^{3}tan x} lim x → 0 x 3 t an x e − 2 x 2 − cos x is equal to
Options A 1 4 \dfrac{1}{4} 4 1 B 1 8 \dfrac{1}{8} 8 1 C 1 12 \dfrac{1}{12} 12 1 D 1 16 \dfrac{1}{16} 16 1 Correct answer C. 1 12 \dfrac{1}{12} 12 1
Step-by-step solution
Using expansion,
lim x → 0 ( 1 + ( − x 2 2 ) + ( x 4 4 ) 2 ! + . . . . ) − ( 1 − x 2 2 ! + x 2 4 ! − . . . . ) x 3 ( x + x 3 3 + . . . . ) \lim_{x \to 0}\dfrac{\left(1 + \left(- \dfrac{x^{2}}{2}\right) + \dfrac{\left(\dfrac{x^{4}}{4}\right)}{2 !} +....\right) - \left(1 - \dfrac{x^{2}}{2 !} + \dfrac{x^{2}}{4 !} -....\right)}{x^{3}\left(x + \dfrac{x^{3}}{3} +....\right)} lim x → 0 x 3 ( x + 3 x 3 + .... ) 1 + ( − 2 x 2 ) + 2 ! ( 4 x 4 ) + .... − ( 1 − 2 ! x 2 + 4 ! x 2 − .... ) lim x → 0 ( 1 − 1 ) + ( − x 2 2 + x 2 2 ) + ( x 4 8 − x 4 24 ) + . . . . x 3 ( x + x 3 3 + . . . . ) \lim_{x \to 0}\dfrac{\left(1 - 1\right) + \left(- \dfrac{x^{2}}{2} + \dfrac{x^{2}}{2}\right) + \left(\dfrac{x^{4}}{8} - \dfrac{x^{4}}{24}\right) +....}{x^{3}\left(x + \dfrac{x^{3}}{3} +....\right)} lim x → 0 x 3 ( x + 3 x 3 + .... ) ( 1 − 1 ) + ( − 2 x 2 + 2 x 2 ) + ( 8 x 4 − 24 x 4 ) + .... ⇒ 1 8 − 1 24 = 1 12 ⇒ \dfrac{1}{8} - \dfrac{1}{24} = \dfrac{1}{12} ⇒ 8 1 − 24 1 = 12 1
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