First slide

Introduction to limits

Question

$lim_{x \to 3} \frac{[x] - 3}{(x - 3)}$ is equal to

Moderate

A

0
B

2
C

3
D

does not exist

Solution

$lim_{x \to 3} \frac{[x] - 3}{(x - 3)}$
Towards the right of r= 3, [r] = 3
[x] - 3 = 0, in the right neighborhood of x = 3
$\Rightarrow lim_{x \to 3 + 0} \frac{[x] - 3}{x - 3} = 0$
Towards the left of x= 3, [x]=2
$\Rightarrow [x] - 3 = - 1$ , in the left neighborhood of x =3
$\Rightarrow lim_{x \to 3 - 0} \frac{[x] - 3}{x - 3} = lim_{x \to 3 - 0} \frac{- 1}{x - 3} = \infty$
Thus, $lim_{x \to 3} \frac{[x] - 3}{(x - 3)}$ does not exist

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