First slide

Introduction to limits

Question

$lim_{x \to \infty} \frac{\log [x]}{x}$ where [x] denotes the greatest integer less than or equal to x, is

Moderate

A

0
B

1
C

-1
D

non-existent

Solution

Let $x = n + k, where 0 < k < 1 . Then, [x] = n$
$\begin{aligned} ∴ lim_{x \to \infty} \frac{\log [x]}{x} = lim_{n \to \infty} \frac{\log n}{n + k} \\ \Rightarrow lim_{x \to \infty} \frac{\log [x]}{x} = lim_{n \to \infty} \frac{1}{n} = 0 \end{aligned}$

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