Question

Evaluate $\underset{x\to 2^{+}}{lim}\frac{[x-2]}{log(x-2)}$, where [.] represents the greatest integer function.

• A. $0$
• B. $-1$
• C. $1$
• D. $\infty$

Answer 1

The correct option is A $0$

$\underset{x\to 2^{+}}{lim}\frac{x-3}{log(x-2)}<\underset{x\to 2^{+}}{lim}\frac{[x-2]}{log(x-2)}<\underset{x\to 2^{+}}{lim}\frac{x-2}{log(x-2)}$

$\Rightarrow \frac{-1}{-\infty }<\underset{x\to 2^{+}}{lim}\frac{[x-2]}{log(x-2)}<\frac{0}{-\infty }$

$\Rightarrow 0<\underset{x\to 2^{+}}{lim}\frac{[x-2]}{log(x-2)}<0$

$\Rightarrow \underset{x\to 2^{+}}{lim}\frac{[x-2]}{log(x-2)}=0$