Question

Solve the limit:

$\lim \limits_{x \to 3^ + } \dfrac x{[x ]}$​

Answer 1

Given:-

$\lim \limits_{x \to 3^ + } \dfrac x{[x ]}$

★SOLUTION:-

$\lim \limits_{x \to 3^ + } \dfrac x{[x ]}$

$\implies \: if \: x \to3 \: \: \: \: then \: \: h \to \: 0$

it means x palace we put (3+h) because we find RHL

$\implies \: lim_{h \to \:0 } \: \frac{3 + h}{[3 + h]}$

greatest intiger function:-

• [2 ]=2

• [3.8]=3

it means

• [3+h]=3

$\implies \: lim_{h \to \: 0} \: \: \: \: \frac{3 + h}{3}$

put limit to we get

$\implies \: \frac{3 + 0}{3} \\ \\ \implies \: \cancel{\frac{3}{3} } \\ \\ \implies \: 1$

______________________________

hope it helps you

Answer 2

Given:

$\lim \limits_{x \to 3^ + } \dfrac x{[x ]}$

Solution:

$= \lim \limits_{h \to 0 } \frac {3 + h}{[3 + h ]}$

$= \lim \limits_{h \to 0 } \frac {(3 + h)}{3 }$

$= \frac{3 + 0}{3}$

$= \frac{3}{3}$

$= 1$