What is \(\lim_{x \to 0} \frac{f(x)}{g(x)}\) equal to?

Calculation:

Given,

The function is \( f(x) = \sin(\lfloor x \rfloor) \), where \(\lfloor x \rfloor \) is the greatest integer function, and g(x) = |x| , the absolute value function.

We are tasked with finding:

\( \lim_{x \to 0} \frac{f(x)}{g(x)} \)

For \( g(x) = |x| \), we know that:

\( \lim_{x \to 0} g(x) = 0 \)

For \( f(x) = \sin(\lfloor x \rfloor) \), we know that:

For \( x \to 0^+ \), \( \lfloor x \rfloor = 0 \), so f(x) = sin(0) = 0 .

For \( x \to 0^- \), \( \lfloor x \rfloor = -1 \), so \( f(x) = \sin(-1) \), which is a nonzero constant.

Evaluating the limit:

For \( x \to 0^+ \), \( \frac{f(x)}{g(x)} = \frac{0}{x} = 0 \)

For \( x \to 0^- \), \( \frac{f(x)}{g(x)} = \frac{\sin(-1)}{-x} \), which becomes undefined as \( x \to 0^- \)because the denominator approaches 0, but the numerator remains a nonzero constant.

∴ Since the left-hand and right-hand limits do not match, the limit does not exist.

The correct answer is Option (4):