If a, b, c and k are non-zero real numbers an | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Given $\lim _{x \rightarrow \infty} x\left(a^{1 / x}+b^{1 / x}+c^{1 / x}-3 k^{1 / x}\right)=0$. Let $y = \frac{1}{x}$. As $x \rightarrow \infty$,$

Vedclass · Accepted Answer

$(abc)^{1/3}$

Vedclass · Answer

(B) Given $\lim _{x \rightarrow \infty} x\left(a^{1 / x}+b^{1 / x}+c^{1 / x}-3 k^{1 / x}\right)=0$. Let $y = \frac{1}{x}$. As $x \rightarrow \infty$,$y \rightarrow 0$. The expression becomes $\lim _{y \rightarrow 0} \frac{a^y+b^y+c^y-3 k^y}{y} = 0$. This can be rewritten as $\lim _{y \rightarrow 0} \left[ \frac{a^y-1}{y} + \frac{b^y-1}{y} + \frac{c^y-1}{y} - 3 \frac{k^y-1}{y} \right] = 0$. Using the standard limit $\lim _{y \rightarrow 0} \frac{a^y-1}{y} = \ln a$,we get: $\ln a + \ln b + \ln c - 3 \ln k = 0$. $\ln (abc) = 3 \ln k$. $\ln (abc) = \ln (k^3)$. Therefore,$k^3 = abc$,which implies $k = (abc)^{1/3}$.

If $a, b, c$ and $k$ are non-zero real numbers and $\lim _{x \rightarrow \infty} x\left(a^{\frac{1}{x}}+b^{\frac{1}{x}}+c^{\frac{1}{x}}-3 k^{\frac{1}{x}}\right)=0$,then $k=$

Similar Questions

$\lim _{x \rightarrow 0}\left(\frac{(x+2 \cos x)^{3}+2(x+2 \cos x)^{2}+3 \sin (x+2 \cos x)}{(x+2)^{3}+2(x+2)^{2}+3 \sin (x+2)}\right)^{\frac{100}{x}}$ is equal to $.....$

$\lim_{x \rightarrow \frac{\pi}{2}} (\tan^{2} x (\sqrt{2 \sin^{2} x + 3 \sin x + 4} - \sqrt{\sin^{2} x + 6 \sin x + 2}))$ is equal to

If $\beta = \lim_{x \rightarrow 0} \frac{e^{x^3} - (1 - x^3)^{1/3} + ((1 - x^2)^{1/2} - 1) \sin x}{x \sin^2 x}$,then the value of $6 \beta$ is

$\lim _{x \rightarrow 0}\left[\tan \left(x+\frac{\pi}{4}\right)\right]^{1 / x}$ is equal to

If $a = \lim_{n \rightarrow \infty} \frac{1+2+3+\ldots+n}{n^2}$ and $b = \lim_{n \rightarrow \infty} \frac{1^2+2^2+3^2+\ldots+n^2}{n^3}$,then

Vedclass Test Series

Exam Paper Generator

Online Exam Module