\( \lim _{z \rightarrow 1+5i} (|z|+z^{2}) \)

To evaluate the limit \( \lim_{z \rightarrow 1+5i} (|z| + z^2) \), we first need to find the values of \( |z| \) and \( z^2 \) as \( z \) approaches \( 1 + 5i \). 1. **Calculate \( |z| \)**: The modulus of \( z \) when \( z = 1 + 5i \) is given by: \[ |z| = |1 + 5i| = \sqrt{1^2 + 5^2} = \sqrt{1 + 25} = \sqrt{26}. \] 2. **Calculate \( z^2 \)**: Next, we compute \( z^2 \) when \( z = 1 + 5i \): \[ z^2 = (1 + 5i)^2 = 1^2 + 2 \cdot 1 \cdot 5i + (5i)^2 = 1 + 10i + 25(-1) = 1 + 10i - 25 = -24 + 10i. \] 3. **Combine the results**: Now we can combine the results of \( |z| \) and \( z^2 \): \[ |z| + z^2 = \sqrt{26} + (-24 + 10i) = \sqrt{26} - 24 + 10i. \] 4. **Take the limit**: As \( z \) approaches \( 1 + 5i \), both \( |z| \) and \( z^2 \) approach their respective values calculated above. Therefore, we have: \[ \lim_{z \rightarrow 1+5i} (|z| + z^2) = \sqrt{26} - 24 + 10i. \] Thus, the final result is: \[ \boxed{\sqrt{26} - 24 + 10i}. \]