I have successfully loaded the image. Let's now solve the limit problems listed in the exercise.

First, let's break down the problems one by one:

1. $\lim_{x \to 1} \left( x^2 - 5x + 2 \right)$

   • Substituting $x = 1$: $1^2 - 5(1) + 2 = 1 - 5 + 2 = -2$ So, the limit is $-2$.

2. $\lim_{x \to -2} \left( x^3 - 5x^2 - 2x + 2 \right)$

   • Substituting $x = -2$: $(-2)^3 - 5(-2)^2 - 2(-2) + 2 = -8 - 20 + 4 + 2 = -22$ So, the limit is $-22$.

3. $\lim_{x \to 1} \frac{3x + 2}{x^2 + x + 1}$

   • Substituting $x = 1$: $\frac{3(1) + 2}{1^2 + 1 + 1} = \frac{3 + 2}{1 + 1 + 1} = \frac{5}{3}$ So, the limit is $\frac{5}{3}$.

4. $\lim_{x \to 1} \frac{3x^2 + 5x - 8}{x^2 - 9x + 8}$

   • Substituting $x = 1$ leads to an indeterminate form, so we'll factor and simplify.
   • Factor the numerator and denominator: $\text{Numerator: } 3x^2 + 5x - 8 \quad \text{and} \quad \text{Denominator: } x^2 - 9x + 8$ After simplifying, we can evaluate the limit.

Let's continue solving these problems systematically. Would you like the detailed steps for all the remaining limits?