Question

\text{الف}) \lim_{x \to 3^+} \frac{x^2 - 9}{x - x}

Explanation:

Evaluate the greatest integer function as \(x \to 3^+\)

$$ \text{For } 3 < x < 4, \quad [x] = 3 $$

Substitute the value of \([x]\) into the limit expression

$$ \lim_{x \to 3^+} \frac{x^2 - 9}{x - [x]} = \lim_{x \to 3^+} \frac{x^2 - 9}{x - 3} $$

Simplify the rational expression and evaluate the limit

$$ \lim_{x \to 3^+} \frac{(x - 3)(x + 3)}{x - 3} = \lim_{x \to 3^+} (x + 3) = 3 + 3 = 6 $$

Answer:

\(6\)