\( \lim _ { + \infty } \frac { \sqrt { x - 1 } - 3 } { x + 3 } \)

Calculate the limit \( \lim_{x\rightarrow +\infty} \frac{\sqrt{x-1}-3}{x+3} \). Evaluate the limit by following steps: - step0: Evaluate using transformations: \(\lim _{x\rightarrow +\infty}\left(\frac{\sqrt{x-1}-3}{x+3}\right)\) - step1: Multiply by the Conjugate: \(\lim _{x\rightarrow +\infty}\left(\frac{\frac{x-10}{\sqrt{x-1}+3}}{x+3}\right)\) - step2: Reduce the fraction: \(\lim _{x\rightarrow +\infty}\left(\frac{x-10}{\left(\sqrt{x-1}+3\right)\left(x+3\right)}\right)\) - step3: Rearrange the terms: \(\lim _{x\rightarrow +\infty}\left(\frac{\left(1-\frac{10}{x}\right)x}{\left(\sqrt{1-\frac{1}{x}}+\frac{3}{x^{\frac{1}{2}}}\right)\left(x^{\frac{1}{2}}+\frac{3}{x^{\frac{1}{2}}}\right)x}\right)\) - step4: Reduce the fraction: \(\lim _{x\rightarrow +\infty}\left(\frac{1-\frac{10}{x}}{\left(\sqrt{1-\frac{1}{x}}+\frac{3}{x^{\frac{1}{2}}}\right)\left(x^{\frac{1}{2}}+\frac{3}{x^{\frac{1}{2}}}\right)}\right)\) - step5: Rewrite the expression: \(\frac{\lim _{x\rightarrow +\infty}\left(1-\frac{10}{x}\right)}{\lim _{x\rightarrow +\infty}\left(\left(\sqrt{1-\frac{1}{x}}+\frac{3}{x^{\frac{1}{2}}}\right)\left(x^{\frac{1}{2}}+\frac{3}{x^{\frac{1}{2}}}\right)\right)}\) - step6: Evaluate: \(\frac{1}{\lim _{x\rightarrow +\infty}\left(\left(\sqrt{1-\frac{1}{x}}+\frac{3}{x^{\frac{1}{2}}}\right)\left(x^{\frac{1}{2}}+\frac{3}{x^{\frac{1}{2}}}\right)\right)}\) - step7: Evaluate: \(\frac{1}{+\infty}\) - step8: Divide the terms: \(0\) The limit of \( \frac{\sqrt{x-1}-3}{x+3} \) as \( x \) approaches positive infinity is 0.