1- Find \( \lim _{x \rightarrow \infty} \sqrt{ } x^{2}+5 x-x \)

To find the limit \( \lim _{x \rightarrow \infty} \sqrt{x^{2}+5x-x} \), we start by simplifying the expression inside the square root: \[ x^{2} + 5x - x = x^{2} + 4x. \] Thus, the limit can be rewritten as: \[ \lim _{x \rightarrow \infty} \sqrt{x^{2} + 4x}. \] Next, we factor out \( x^{2} \) from the square root: \[ \sqrt{x^{2} + 4x} = \sqrt{x^{2}(1 + \frac{4}{x})} = x\sqrt{1 + \frac{4}{x}}. \] Now, as \( x \rightarrow \infty \), the term \( \frac{4}{x} \) approaches 0. Therefore, the expression simplifies to: \[ x\sqrt{1 + \frac{4}{x}} \rightarrow x\sqrt{1 + 0} = x. \] Finally, we evaluate the limit: \[ \lim _{x \rightarrow \infty} x = \infty. \] Thus, the final answer is: \[ \lim _{x \rightarrow \infty} \sqrt{x^{2}+5x-x} = \infty. \]