The limit you provided is:

$\lim_{x \to -2^-} \frac{2x^2 + 7x - 4}{x^2 + x - 2}$

To solve this, let's follow these steps:

1. Factor the numerator and denominator

We will start by factoring both the numerator and denominator, if possible.

Numerator:

$2x^2 + 7x - 4$

To factor, we need two numbers that multiply to $2 \times (-4) = -8$ and add to $7$. The factors are $8$ and $-1$. So we can rewrite the middle term:

$2x^2 + 8x - x - 4 = 2x(x + 4) - 1(x + 4) = (2x - 1)(x + 4)$

Denominator:

$x^2 + x - 2$

We need two numbers that multiply to $-2$ and add to $1$. The factors are $2$ and $-1$:

$x^2 + x - 2 = (x - 1)(x + 2)$

2. Rewrite the expression with factored terms

Now we have:

$\frac{(2x - 1)(x + 4)}{(x - 1)(x + 2)}$

3. Evaluate the limit as $x \to -2^-$

Substitute $x = -2$ into the factored expression:

$\frac{(2(-2) - 1)((-2) + 4)}{((-2) - 1)((-2) + 2)}$

This simplifies to:

$\frac{(-4 - 1)(-2 + 4)}{(-2 - 1)(-2 + 2)} = \frac{(-5)(2)}{(-3)(0)}$

The denominator becomes $0$, which means the function is undefined at $x = -2$, and we need to analyze the behavior of the function as $x \to -2^-$ (from the left).

• As $x \to -2^-$, $x + 2$ approaches $0^-$ (a small negative value).
• The numerator will still approach $(-5)(2) = -10$.
• The denominator will approach a small negative value because $x - 1$ is negative and $x + 2$ approaches $0^-$.

Thus, as $x \to -2^-$, the expression behaves like:

$\frac{-10}{\text{small negative number}} \to +\infty$

Conclusion:

The limit as $x \to -2^-$ of the given function is $+\infty$.

Would you like further details or clarifications?

Follow-up Questions:

1. How do one-sided limits differ from two-sided limits?
2. Why does the denominator becoming zero affect the limit?
3. How can we use L'Hopital's Rule in limit problems involving indeterminate forms?
4. Can the function in the example be continuous at $x = -2$?
5. What are the implications if we approach $x = -2$ from the right side?

Tip:

Always check if factoring helps simplify a limit expression before applying advanced techniques like L'Hopital's Rule!