First slide

Introduction to limits

Question

Let $f (x) = [x] =$ Greatest integer less than or equal to $x$ and $k$ be an integer. Then, which one of the following is not correct?

Moderate

A

$lim_{x \to k^{-}} f (x) = k - 1$
B

$lim_{x \to k^{+}} f (x) = k$
C

$lim_{x \to k} f (x)$ exist
D

$lim_{x \to k} f (x)$ does not exist

Solution

We have, $f (x) = [x]$
$∴$ (LHS at x =k)
$\begin{array}{l} = lim_{x \to k^{-}} f (x) = lim_{h \to 0} f (k - h) = lim_{h \to 0} [k - h] \\ = lim_{h \to 0} k - 1 = k - 1 [∵ k - 1 < k - h < k ∴ [k - h] = k - 1] \\ ∴ (RHL at x = k) \\ = lim_{x \to k^{+}} f (x) = lim_{h \to 0} f (k + h) = lim_{h \to 0} [k + h] \\ = lim_{h \to 0} k = k \end{array}$
Clearly , $lim_{x \to k^{-}} f (x) \neq lim_{x \to k^{+}} f (x) .$ So , $lim_{x \to k} f (x)$ does not exist .

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