& If p,q,r,s > 0, then _ x → ∞ ( p(1)/(x) + q(1)/(x) + r(1)/(x) + s(1)/(x) 4 )^ 3 x is equal to

Correct answer: D. (pqrs) (3)/(4) Consider the given expression, _ x → ∞ ( p 1 x + q 1 x + r 1 x + s 1 x 4 ) ^ 3 x [where, p,q,r,s > 0] Put, 1 x = z , so that x → ∞ ⇒ z → 0 = _ z → 0 ( p^ z + q^ z + r^ z + s^ z 4 ) 3 z which is in the form 1 ^ ∞ . =e ^ _ z → 0 ( p^ z + q^ z + r^ z + s^ z 4 - 1)(3)/(z) = e^ _ z → 0 ( p^ z - 1 z + q^ z - 1 z + r^ z - 1 z + s^ z - 1 z ) (3)/(4) = e^ (3)/(4) ( _ e p + _ e q + _ e r + _ e s) = e^ (3)/(4) _ e (pqrs) = (pqrs)(3)/(4)

NTA Abhyas 2020

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

NTA Abhyas 2020 · Mathematics · Limits

& If p,q,r,s > 0, then _ x → ∞ ( p(1)/(x) + q(1)/(x) + r(1)/(x) +...

& If

\text{p,}\text{q,}\text{r,}\text{s} > 0,

then

\lim_{\text{x} \to \infty}\left(\dfrac{\text{p}\dfrac{1}{x} + \text{q}\dfrac{1}{x} + \text{r}\dfrac{1}{x} + \text{s}\dfrac{1}{x}}{4}\right)^{3}\text{x}

is equal to

Options

A $\text{pqrs}$
B{ $\left(\text{pqrs}\right)}^{3}$
C $\left(\text{pqrs}\right) \dfrac{3}{2}$
D $\left(\text{pqrs}\right) \dfrac{3}{4}$

Correct answer

D. $\left(\text{pqrs}\right) \dfrac{3}{4}$

Step-by-step solution

Consider the given expression,

\lim_{\text{x} \to \infty} {\left(\dfrac{\text{p} \dfrac{1}{\text{x}} + \text{q} \dfrac{1}{\text{x}} + \text{r} \dfrac{1}{\text{x}} + \text{s} \dfrac{1}{\text{x}}}{4}\right)}^{3 \text{x}}

[where,

\text{p,}\text{q,}\text{r,}\text{s > 0]}

Put,

\dfrac{1}{\text{x}} = \text{z}

, so that

\text{x} \to \infty ⇒ \text{z} \to 0

= \lim_{\text{z} \to 0} \left(\dfrac{p^{z} + q^{z} + r^{z} + s^{z}}{4}\right) \dfrac{3}{\text{z}}

which is in the form

{1}^{∞}

{=e}^{\lim_{z \to 0}\left(\dfrac{p^{z} + q^{z} + r^{z} + s^{z}}{4} - 1\right)\dfrac{3}{z}}

= e^{\lim_{\text{z} \to 0} \left(\dfrac{p^{z} - 1}{\text{z}} + \dfrac{q^{z} - 1}{\text{z}} + \dfrac{r^{z} - 1}{\text{z}} + \dfrac{s^{z} - 1}{\text{z}}\right) \dfrac{3}{4}}

= e^{\dfrac{3}{4} \left(\log_{\mathrm{e}} \text{p} + \log_{\mathrm{e}} \text{q} + \log_{\mathrm{e}} \text{r} + \log_{\mathrm{e}} \text{s}\right)}

= e^{\dfrac{3}{4}\log_{\mathrm{e}}\left(\text{pqrs}\right)} = \left(\text{pqrs}\right)\dfrac{3}{4}

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