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The image presents a set of mathematical problems focused on the continuity of functions at specific points. Here's a breakdown of what the tasks ask for:

1. **First question: Continuity study for different functions at given points:**

   You are asked to determine the continuity of several functions at specified points \( x_0 \).

   - **a)** \( f(x) = \frac{-2x^2 - x + 1}{x + 1} \) for \( x \neq -1 \), with \( x_0 = -1 \), and \( f(-1) = 1 \).
   - **b)** \( f(x) = \frac{x + \tan(2x)}{\sin(3x)} \) for \( x \neq 0 \), with \( x_0 = 0 \), and \( f(0) = 1 \).
   - **c)** \( f(x) = \frac{\sqrt{x^2 + x + 2} - 2}{x - 1} \) for \( x \neq 1 \), with \( x_0 = 1 \), and \( f(1) = \frac{3}{4} \).
   - **d)** \( f(x) = \frac{\sin(x)}{\sqrt{x+1} - 1} \) for \( x \neq 0 \), with \( x_0 = 0 \), and \( f(0) = 2 \).

   For each part, you need to:
   - Determine the limit of the function as \( x \) approaches the given point \( x_0 \).
   - Compare this limit to the value of the function at the point (if it exists) to verify if the function is continuous.

2. **Second question: Continuity of function \( g(x) \) at \( x_0 = 2 \)**

   You are asked to find the value of \( \lambda \) such that the function \( g(x) = \frac{x^3 - 2x^2 - x + 2}{x^2 - 4} \), with \( g(2) = \lambda \), is continuous at \( x_0 = 2 \).

Would you like me to walk you through the detailed solutions for each part of the question? Let me know if you'd like to start with a specific part or if you have a question on continuity.

Dans chacun des cas suivants, étudier la continuité de la fonction f au point x_0.

This problem involves determining the continuity of various functions, including algebraic and trigonometric types, at specific points. Explore the limits, continuity definitions, and solve step-by-step to determine if these functions are continuous at the given points.

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