lim _(narrow infty )((x^n-2)/(x^n)-x)^n+1

To find the limit of the given expression as \( n \) approaches infinity, we can simplify the expression first.<br /><br />Given expression:<br />\[ \lim_{n \to \infty} \left( \frac{x^{n-2}}{x^n - x} \right)^{n+1} \]<br /><br />First, let's rewrite the fraction inside the limit:<br />\[ \frac{x^{n-2}}{x^n - x} = \frac{x^{n-2}}{x^n (1 - \frac{x}{x^n})} = \frac{x^{n-2}}{x^n (1 - \frac{1}{x^{n-1}})} \]<br /><br />As \( n \to \infty \), \(\frac{1}{x^{n-1}} \to 0\), so:<br />\[ \frac{x^{n-2}}{x^n (1 - \frac{1}{x^{n-1}})} \approx \frac{x^{n-2}}{x^n} = \frac{1}{x^2} \]<br /><br />Now, substitute this back into the original limit:<br />\[ \lim_{n \to \infty} \left( \frac{1}{x^2} \right)^{n+1} = \left( \frac{1}{x^2} \right)^{\infty} \]<br /><br />Since \(\frac{1}{x^2}\) is a constant, raising it to the power of infinity will result in:<br />\[ \left( \frac{1}{x^2} \right)^{\infty} = 0 \]<br /><br />Therefore, the limit is:<br />\[ \boxed{0} \]