I see that you have uploaded an image with mathematical problems covering topics like limits, continuity, and applications of limits to real-life scenarios. Let me assist you with the solutions for these problems step by step.

I. Definição intuitiva do limite

• These are graphical questions. Based on the graph of the function $f(x)$, you need to determine the limit as $x$ approaches specific points. Since I don't have access to the visual part of the graph, if you describe the function or provide the key features of the graph, I can help solve them.

II. Cálculo de limites de funções

Let’s solve some of these limit problems:

1. $\lim_{x \to -2} \left( x^3 + 8x^2 - 1 \right)$

   • Substituting $x = -2$ directly gives: $(-2)^3 + 8(-2)^2 - 1 = -8 + 8(4) - 1 = -8 + 32 - 1 = 23$ Therefore, the limit is 23.

2. $\lim_{x \to 2} \cos(x^2 - 5x + 6)$

   • First, simplify the expression inside the cosine function: $x^2 - 5x + 6 = (x - 2)(x - 3)$ So when $x = 2$, the expression becomes: $\cos(2^2 - 5(2) + 6) = \cos(4 - 10 + 6) = \cos(0) = 1$ Therefore, the limit is 1.

3. $\lim_{x \to 7} \left( 3x + 2 \right)^{\frac{2}{3}}$

   • Substitute $x = 7$: $\left( 3(7) + 2 \right)^{\frac{2}{3}} = \left( 21 + 2 \right)^{\frac{2}{3}} = 23^{\frac{2}{3}}$ This is the simplified form of the limit, approximately equal to 6.32.

Would you like to continue solving more of these limits, or proceed with the later sections?

5 Related Questions:

1. What is the general method for calculating limits of trigonometric functions?
2. How do we approach solving limits when direct substitution leads to an indeterminate form?
3. What are the rules for calculating limits involving square roots?
4. How can we graphically interpret one-sided limits?
5. What is the behavior of a function as $x$ approaches infinity?

Tip:

When calculating limits, always try direct substitution first. If it results in an indeterminate form, consider using algebraic simplifications or L'Hopital's Rule.