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The expression you have uploaded is:

\[
\lim_{{x \to 3^-}} \left( \frac{5x^2 - 2}{5x^2 - 13x - 6} - \frac{x}{x - 3} \right)
\]

Let's evaluate this limit step by step.

### Step 1: Analyze the expression.
We have two terms within the parentheses, and we'll need to combine them before attempting to evaluate the limit. The terms are:
- \(\frac{5x^2 - 2}{5x^2 - 13x - 6}\)
- \(\frac{x}{x - 3}\)

We'll start by finding a common denominator so that we can combine them.

### Step 2: Factor the denominator of the first fraction.
The denominator \(5x^2 - 13x - 6\) can be factored as:
\[
5x^2 - 13x - 6 = (5x + 2)(x - 3)
\]

### Step 3: Combine the fractions.
Now that we have the factored form, the common denominator between the two fractions is \((5x + 2)(x - 3)\). We can rewrite the expression as:
\[
\frac{5x^2 - 2}{(5x + 2)(x - 3)} - \frac{x}{x - 3}
\]
Rewrite the second term so that it has the same denominator:
\[
\frac{5x^2 - 2}{(5x + 2)(x - 3)} - \frac{x(5x + 2)}{(5x + 2)(x - 3)}
\]

### Step 4: Combine the numerators.
Now that we have the same denominator, we can subtract the numerators:
\[
\frac{(5x^2 - 2) - x(5x + 2)}{(5x + 2)(x - 3)}
\]

Simplify the numerator:
\[
(5x^2 - 2) - x(5x + 2) = 5x^2 - 2 - 5x^2 - 2x = -2 - 2x
\]

Thus, the expression becomes:
\[
\frac{-2x - 2}{(5x + 2)(x - 3)}
\]
We can factor the numerator as:
\[
\frac{-2(x + 1)}{(5x + 2)(x - 3)}
\]

### Step 5: Evaluate the limit as \(x \to 3^-\).
Now, substitute \(x = 3\) into the simplified expression. However, notice that the denominator will become 0, specifically due to the \((x - 3)\) term. Let's check the sign of the result from both the numerator and denominator as \(x \to 3^-\).

- As \(x \to 3^-\), \(x - 3\) approaches 0 from the negative side.
- As \(x \to 3^-\), \(5x + 2\) approaches \(5(3) + 2 = 15 + 2 = 17\), which is positive.
- The numerator \(-2(x + 1)\) approaches \(-2(3 + 1) = -2(4) = -8\), which is negative.

Thus, the expression behaves like:
\[
\frac{\text{negative number}}{\text{positive number} \times \text{negative number}} = \text{positive}
\]

The overall result will tend toward \(+\infty\).

### Final Answer:
\[
\lim_{{x \to 3^-}} \left( \frac{5x^2 - 2}{5x^2 - 13x - 6} - \frac{x}{x - 3} \right) = +\infty
\]

Would you like more details on any of the steps or further clarification?

Here are 5 related questions you might explore:
1. How does factoring polynomials help in simplifying limits?
2. Why does division by zero lead to infinite limits in certain cases?
3. What does it mean when the limit tends to infinity from the left-hand side (\(x \to 3^-\))?
4. How can you use L'Hopital's Rule for indeterminate forms like this?
5. What are the types of discontinuities, and how can limits help identify them?

**Tip:** When encountering indeterminate forms like \(\frac{0}{0}\) in limits, try factoring or simplifying the expression first before using advanced techniques like L'Hopital's Rule.

Evaluate the limit: lim_{x -> 3^-} [(5x^2 - 2)/(5x^2 - 13x - 6) - x/(x - 3)]

Learn how to solve the limit lim_{x -> 3^-} [(5x^2 - 2)/(5x^2 - 13x - 6) - x/(x - 3)] by factoring, simplifying rational expressions, and analyzing asymptotic behavior. This step-by-step guide helps in understanding limits involving division by zero.

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