lim _(n arrow infty)(sqrt(n^2)+3 cdot 5 cdot n-5)

To find the limit of the given expression as \( n \) approaches infinity, we can simplify the expression inside the square root.<br /><br />First, let's rewrite the expression inside the square root:<br /><br />\[ \sqrt{n^2 + 3 \cdot 5 \cdot n - 5} \]<br /><br />This simplifies to:<br /><br />\[ \sqrt{n^2 + 15n - 5} \]<br /><br />Next, we can factor out \( n^2 \) from the terms inside the square root:<br /><br />\[ \sqrt{n^2(1 + \frac{15}{n} - \frac{5}{n^2})} \]<br /><br />Now, we can take the square root of \( n^2 \) outside the square root:<br /><br />\[ n \sqrt{1 + \frac{15}{n} - \frac{5}{n^2}} \]<br /><br />As \( n \) approaches infinity, the terms \( \frac{15}{n} \) and \( \frac{5}{n^2} \) approach 0. Therefore, the expression simplifies to:<br /><br />\[ n \sqrt{1 + 0 - 0} = n \sqrt{1} = n \]<br /><br />So, the limit of the given expression as \( n \) approaches infinity is \( n \).