Let f be a function satisfying the condition - fxy-fxy-fyx - x- y - 0- If fx is differentiable and f1-1- then the value of limx-x fx is-

The correct option is A 1 λ f(xy)=f(x)y+f(y)x Let y=1 f(1)=1 or f(y)=1 λ f(x)=f(x)1+f(y)x λ f(x) f(x)=1x (λ 1)f(x)=1 ...

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Standard XII

Mathematics

Definition of Ellipse

Let f be a fu...

Question

Let f be a function satisfying the condition $λ f (x y) = \frac{f (x)}{y} + \frac{f (y)}{x}$ $\forall$ x, y $> 0$ . If $f (x)$ is differentiable and $f (1) = 1$ , then the value of $lim x \to \infty x$ $f (x)$ is?

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Solution

The correct option is A $1$
$λ f (x y) = \frac{f (x)}{y} + \frac{f (y)}{x}$

Let $y = 1$ $f (1) = 1$ or $f (y) = 1$

$λ f (x) = \frac{f (x)}{1} + \frac{f (y)}{x}$

$λ f (x) - f (x) = \frac{1}{x}$

$(λ - 1) f (x) = \frac{1}{x}$

Only $f (x) = \frac{1}{x}; (λ = 2)$

satisfies equation

hence , $f (x) = \frac{1}{x}$

$lim x \to \infty x f (x)$

$lim x \to \infty x \times \frac{1}{x} = 1$

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Let f be a function satisfying the condition λf(xy)=f(x)y+f(y)x ∀ x, y >0. If f(x) is differentiable and f(1)=1, then the value of limx→∞x f(x) is?

The correct option is A 1λf(xy)=f(x)y+f(y)xLet y=1 f(1)=1 or f(y)=1λf(x)=f(x)1+f(y)xλf(x)−f(x)=1x(λ−1)f(x)=1xOnly f(x)=1x;(λ=2)satisfies equationhence , f(x)=1xlimx→∞xf(x)limx→∞x×1x=1

Let f be a function satisfying the condition $λ f (x y) = \frac{f (x)}{y} + \frac{f (y)}{x}$ $\forall$ x, y $> 0$ . If $f (x)$ is differentiable and $f (1) = 1$ , then the value of $lim x \to \infty x$ $f (x)$ is?