This document contains several exercises on functions, continuity, limits, and graphical analysis. I'll summarize the key tasks for each exercise.

Exercise N°1

• Given: A piecewise function $f(x)$ defined over real numbers $\mathbb{R}$. \frac{\sqrt{x^2 + 3} - x}{3x - 4} & \text{if } x \leq 1 \\ x^2 - 5 & \text{if } x > 1 \end{cases}$$
• Tasks:
  1. Show continuity: Prove that $f(x)$ is continuous at $x = 1$.
  2. Calculate limits: Compute the following limits:
     • $\lim_{x \to -\infty} f(x)$
     • $\lim_{x \to +\infty} f(x)$
     • $\lim_{x \to 0} \frac{f(x)}{x^2}$

Exercise N°2

• Given: The graph of a function $f$.
• Tasks:
  1. Determine the domain of $f$.
  2. Identify whether $f$ is even.
  3. Evaluate $f$ at specific points.
     • $f(-3)$
     • $f(-1)$
     • $f(3)$
     • $f(-1)$
  4. Calculate the limits $\lim_{x \to -3} f(x)$ and $\lim_{x \to 3} f(x)$.

Exercise N°3

• Given: The graph of a function $f$ defined over $[-1, 8]$.
• Tasks: Using the graph:
  1. Determine $f(0)$, $f(-1)$.
  2. Calculate several limits:
     • $\lim_{x \to 1^-} f(x)$
     • $\lim_{x \to 1^+} f(x)$
     • $\lim_{x \to +\infty} f(x)$
  3. Evaluate the overall limit $\lim_{x \to -1} f(x)$.
  4. Find the limit as $x \to 0$ of $\frac{f(x)}{x}$.

Exercise N°4

• Given: A curve $\mathcal{C}_f$ representing a continuous function.
• Additional Information:
  • The curve has a parabolic branch.
  • There is an asymptote with an equation $x = 1$.
  • The line $y = x$ is an asymptote at $+\infty$.
• Tasks: Using the graph:
  1. Find $f(0)$, $f(-1)$.
  2. Determine specific limits:
     • $\lim_{x \to -\infty} f(x)$
     • $\lim_{x \to 1^-} f(x)$
     • $\lim_{x \to 1^+} f(x)$
     • $\lim_{x \to +\infty} f(x)$
  3. Find the derivative $f'(x)$ at specific points.
  4. Calculate $\lim_{x \to 0} f(x)$.

Would you like a detailed solution for any specific exercise?

Follow-up Questions

1. How can you prove the continuity of a piecewise function at a particular point?
2. What steps are involved in determining if a function is even or odd?
3. How do limits at infinity provide insights into the behavior of a function?
4. What does it mean for a function to have an asymptote, and how is it identified graphically?
5. Why is it essential to analyze left-hand and right-hand limits at points of discontinuity?

Tip:

When analyzing a graph, always look for intercepts, asymptotes, and critical points first to guide your interpretation of the function's behavior.