First slide

Introduction to limits

Question

$lim_{x \to \infty} \frac{\log x}{[x]}$ where [·] denotes the greatest integer function, is

Moderate

A

0
B

1
C

-1
D

non-of these

Solution

We have,
$\begin{array}{l} x - 1 < [x] \leq x for all x \in R \\ \Rightarrow \frac{1}{x} \leq \frac{1}{[x]} < \frac{1}{x - 1} for ail x \in R - {0, 1} \\ \Rightarrow \frac{\log x}{x} \leq - \frac{\log x}{[x]} < \frac{\log x}{x - 1} [∵ \log x > 0 as x \to x] \\ \Rightarrow lim_{x \to \infty} \frac{\log x}{x} \leq lim_{x \to \infty} \frac{\log x}{[x]} < lim_{x \leq \infty} \frac{\log x}{x - 1} \\ \Rightarrow lim_{x \to \infty} \frac{\log x}{[x]} = 0 [∵ lim_{x \to \infty} \frac{\log x}{x} = lim_{x \to \infty} \frac{\log x}{x - 1} = i \end{array}$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App