Let f(x)=( ln (x2+ex)/ ln (x4+e2 x)), then value of undersetx arrow ∞ textLim((1/f(x)))f(x) is equal to

Question

Let f(x)=( ln (x2+ex)/ ln (x4+e2 x)), then value of undersetx arrow ∞ textLim((1/f(x)))f(x) is equal to (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

undersetx arrow ∞ textLim f(x)= undersetx arrow ∞ textLim(x+ ln (e-x x2+1)/2 x+ ln (e-x x4+1))=(1/2) ∵ undersetx arrow ∞ textLime-x x4= undersetx arrow ∞ textLim (x4/ex)=0 ∴ =2(1/2)=√2

Q. Let $f (x) = \frac{l n ( x ^{2} + e ^{x} )}{l n ( x ^{4} + e ^{2 x} )}$ , then value of $x \to \infty Lim (\frac{1}{f ( x )})^{f (x)}$ is equal to

$x \to \infty Lim f (x) = x \to \infty Lim \frac{x + l n ( e ^{- x} x ^{2} + 1 )}{2 x + l n ( e ^{- x} x ^{4} + 1 )} = \frac{1}{2}$
$∵ x \to \infty Lim e^{- x} x^{4} = x \to \infty Lim \frac{x ^{4}}{e ^{x}} = 0$
$∴ = 2^{\frac{1}{2}} = 2$

Q. Let f(x)=ln(x4+e2x)ln(x2+ex)​, then value of x→∞Lim​(f(x)1​)f(x) is equal to

x→∞Lim​f(x)=x→∞Lim​2x+ln(e−xx4+1)x+ln(e−xx2+1)​=21​ ∵x→∞Lim​e−xx4=x→∞Lim​exx4​=0 ∴=221​=2​

Q. Let $f (x) = \frac{l n ( x ^{2} + e ^{x} )}{l n ( x ^{4} + e ^{2 x} )}$ , then value of $x \to \infty Lim (\frac{1}{f ( x )})^{f (x)}$ is equal to

$x \to \infty Lim f (x) = x \to \infty Lim \frac{x + l n ( e ^{- x} x ^{2} + 1 )}{2 x + l n ( e ^{- x} x ^{4} + 1 )} = \frac{1}{2}$
$∵ x \to \infty Lim e^{- x} x^{4} = x \to \infty Lim \frac{x ^{4}}{e ^{x}} = 0$
$∴ = 2^{\frac{1}{2}} = 2$