The value of _ x → 0 ^ - 2 ^ (1)/(x) + 2 ^ (3)/(x) 3( 2 ^ (1)/(x) ) + 5( 2 ^ (3)/(x) ) is

Correct answer: A. (1)/(3) _ x → 0 ^ - 2 ^ (1)/(x) \ 1 + 2 ^ (2)/(x) \ 2 ^ (1)/(x) \ 3 + 5 ⋅ 2 ^ (2)/(x) \ ⇒ _ x → 0 ^ - 1 + 2 ^ (1)/(x) 3 + 5( 2 ^ (2)/(x) ) = (1 + 0)/(3 + 0) = (1)/(3)

NTA Abhyas 2020

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

NTA Abhyas 2020 · Mathematics · Limits

The value of _ x → 0 ^ - 2 ^ (1)/(x) + 2 ^ (3)/(x) 3( 2 ^ (1)/(x) )...

The value of

\lim_{x \to {0}^{-}}\dfrac{{2}^{\dfrac{1}{x}} + {2}^{\dfrac{3}{x}}}{3\left({2}^{\dfrac{1}{x}}\right) + 5\left({2}^{\dfrac{3}{x}}\right)}

Options

A $\dfrac{1}{3}$
B $\dfrac{1}{5}$
C $1$
D $\dfrac{1}{4}$

Correct answer

A. $\dfrac{1}{3}$

Step-by-step solution

\lim_{x \to {0}^{-}}\dfrac{{2}^{\dfrac{1}{x}}\left\{1 + {2}^{\dfrac{2}{x}}\right\}}{{2}^{\dfrac{1}{x}}\left\{3 + 5 ⋅ {2}^{\dfrac{2}{x}}\right\}} ⇒ \lim_{x \to {0}^{-}}\dfrac{1 + {2}^{\dfrac{1}{x}}}{3 + 5\left({2}^{\dfrac{2}{x}}\right)}

= \dfrac{1 + 0}{3 + 0} = \dfrac{1}{3}

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