Evaluate $$ \lim _{x \rightarrow 0^{-}} \frac{5 x-4|x|}{3 x} $$

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To evaluate the limit
$$
\lim _{x \rightarrow 0^{-}} \frac{5 x - 4|x|}{3 x},
$$
we analyze the behavior of the expression as $$ x $$ approaches $$ 0 $$ from the negative side ($$ x \rightarrow 0^{-} $$).

1. **Determine the Expression for $$ |x| $$ When $$ x < 0 $$:**
   Since $$ x $$ is approaching $$ 0 $$ from the negative side, $$ x < 0 $$. Therefore, the absolute value of $$ x $$ is:
   $$
   |x| = -x
   $$

2. **Substitute $$ |x| $$ into the Original Expression:**
   $$
   \frac{5x - 4|x|}{3x} = \frac{5x - 4(-x)}{3x} = \frac{5x + 4x}{3x} = \frac{9x}{3x}
   $$

3. **Simplify the Expression:**
   For $$ x \neq 0 $$, we can cancel out $$ x $$ from the numerator and the denominator:
   $$
   \frac{9x}{3x} = \frac{9}{3} = 3
   $$

4. **Evaluate the Limit:**
   Since the expression simplifies to a constant (3) for all $$ x \neq 0 $$, the limit as $$ x $$ approaches $$ 0 $$ from the negative side is:
   $$
   \lim _{x \rightarrow 0^{-}} \frac{5 x - 4|x|}{3 x} = 3
   $$

**Answer:** 3
The limit is 3.

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