displaystyle lim x arrow ∞ ( log [x]/x), where [.] denotes greatest integral function, is

Question

displaystyle lim x arrow ∞ ( log [x]/x), where [.] denotes greatest integral function, is (A) 0 (B) 1 (C) -1 (D) none of these

Tardigrade Admin · Accepted Answer

displaystyle lim x arrow ∞ ( log [x]/x)= lim x arrow ∞ ( log x/x) (∵ when x arrow ∞,[x] ≈ x ) = displaystyle lim x arrow ∞ ((1/x)/1)=0 (Applying L' Hospital's rule)

Q. $x \to \infty lim \frac{lo g [ x ]}{x}$ , where [.] denotes greatest integral function, is

0

1

-1

none of these

$x \to \infty lim \frac{lo g [ x ]}{x} = x \to \infty lim \frac{lo g x}{x}$
$(∵$ when $x \to \infty, [x] \approx x$ )
$= x \to \infty lim \frac{\frac{1}{x}}{1} = 0$
(Applying L' Hospital's rule)

Q. x→∞lim​xlog[x]​, where [.] denotes greatest integral function, is

0

1

-1

none of these

x→∞lim​xlog[x]​=x→∞lim​xlogx​ (∵ when x→∞,[x]≈x ) =x→∞lim​1x1​​=0 (Applying L' Hospital's rule)

Q. $x \to \infty lim \frac{lo g [ x ]}{x}$ , where [.] denotes greatest integral function, is

$x \to \infty lim \frac{lo g [ x ]}{x} = x \to \infty lim \frac{lo g x}{x}$
$(∵$ when $x \to \infty, [x] \approx x$ )
$= x \to \infty lim \frac{\frac{1}{x}}{1} = 0$
(Applying L' Hospital's rule)