What \(\lim_{x \to 0} {f(x) g(x)}\) is  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:

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

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\(x \to 0^+ \),\(f(x)g(x) = 0 \times x = 0 \)

For\(x \to 0^- \), \(f(x)g(x) = \sin(-1) \times (-x) \), which approaches 0 as \(x \to 0^- .\).

∴ The value of \(\lim_{x \to 0} f(x) g(x) \)is 0.

The correct answer is Option (2).