lim _(x arrow 9) (x^m-a^m)/(x^n)-a^(n)

To evaluate the limit \( \lim_{x \rightarrow 9} \frac{x^m - a^m}{x^n - a^n} \), we can use L'Hôpital's Rule if necessary. However, let's first check if direct substitution works.<br /><br />Substitute \( x = 9 \) into the expression:<br /><br />\[ \frac{9^m - a^m}{9^n - a^n} \]<br /><br />If this expression is indeterminate (e.g., \( \frac{0}{0} \)), we can apply L'Hôpital's Rule, which states that if the limit of the ratio of two functions results in an indeterminate form, then the limit of the ratio of their derivatives is the same.<br /><br />Let's find the derivatives of the numerator and the denominator with respect to \( x \):<br /><br />1. The derivative of the numerator \( x^m - a^m \) with respect to \( x \) is \( mx^{m-1} \).<br />2. The derivative of the denominator \( x^n - a^n \) with respect to \( x \) is \( nx^{n-1} \).<br /><br />Now, apply L'Hôpital's Rule:<br /><br />\[ \lim_{x \rightarrow 9} \frac{x^m - a^m}{x^n - a^n} = \lim_{x \rightarrow 9} \frac{mx^{m-1} \cdot 1}{nx^{n-1} \cdot 1} = \lim_{x \rightarrow 9} \frac{m x^{m-1}}{n x^{n-1}} \]<br /><br />Simplify the expression:<br /><br />\[ \lim_{x \rightarrow 9} \frac{m x^{m-1}}{n x^{n-1}} = \lim_{x \rightarrow 9} \frac{m}{n} x^{m-n} \]<br /><br />Since \( m \neq n \), \( x^{m-n} \) does not go to zero as \( x \) approaches 9. Therefore, the limit simplifies to:<br /><br />\[ \lim_{x \rightarrow 9} \frac{m}{n} x^{m-n} = \frac{m}{n} \cdot 9^{m-n} \]<br /><br />Thus, the final answer is:<br /><br />\[ \lim_{x \rightarrow 9} \frac{x^m - a^m}{x^n - a^n} = \frac{m}{n} \cdot 9^{m-n} \]