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.