1. **Problem 1: Find the limit** $$\lim_{x \to 1} \frac{x^2 - 1}{x^2 + 1}$$ 2. Substitute $x=1$: numerator is $1^2 - 1 = 0$, denominator is $1^2 + 1 = 2$, so limit is $0/2=0$ directly. 3. **Problem 2: Given matrices** $$A = \begin{bmatrix} 1 & 1 & -2 \\ -1 & 2 & 1 \\ 0 & 1 & -1 \end{bmatrix}$$ Find eigenvalues and eigenvectors: - Compute characteristic polynomial $\det(A - \lambda I) = 0$. - Solve for eigenvalues $\lambda$. - For each $\lambda$, solve $(A - \lambda I)\mathbf{v}=0$ for eigenvectors $\mathbf{v}$. 4. **Problem 3: Given matrix** $$A = \begin{bmatrix} 2 & -1 & 2 \\ -1 & 2 & -1 \\ 1 & -1 & 2 \end{bmatrix}$$ Find $A^{-1}$: - Calculate determinant $\det(A)$. - Find matrix of cofactors. - Transpose cofactor matrix to get adjugate matrix. - Compute $A^{-1} = \frac{1}{\det(A)} \mathrm{adj}(A)$. ### Final answers summary: - $$\lim_{x \to 1} \frac{x^2 - 1}{x^2 + 1} = 0$$ - Eigenvalues and eigenvectors require solving $\det(A - \lambda I) = 0$ and $(A - \lambda I)\mathbf{v}=0$ for given matrix in problem 2. - $A^{-1}$ for problem 3 is found by formula involving determinant and adjugate as explained.