91Ó°ÊÓ

Rewrite the given expression in a simplified form, so it'll be easier to evaluate the limit. The expression can be rewritten as: \[\lim _{x \rightarrow 1} \left\\{\frac{10}{1-x^{10}} - \frac{9}{1-x^{9}}\right\\}\]

To further simplify the expression, we'll combine the fractions by finding a common denominator, which in this case is \((1-x^{10})(1-x^9)\). Rewrite the expression with the common denominator: \[\lim _{x \rightarrow 1} \left\\{\frac{10(1-x^9) - 9(1-x^{10})}{(1-x^{10})(1-x^9)} \right\\}\] Now, expand the numerator: \[\lim _{x \rightarrow 1} \left\\{\frac{10 - 10x^9 - 9 +9x^{10}}{(1-x^{10})(1-x^9)} \right\\}\] Next, simplify the numerator by combining like terms: \[\lim _{x \rightarrow 1} \left\\{\frac{1 - 10x^9 + 9x^{10}}{(1-x^{10})(1-x^9)} \right\\}\]

The numerator seems to have a common factor that we can factor out and simplify: \[\lim _{x \rightarrow 1} \left\\{\frac{1x^9(1 - 10 + 9x)}{(1-x^{10})(1-x^9)} \right\\}\] Make sure to distribute the x^9: \[\lim _{x \rightarrow 1} \left\\{\frac{x^9(1 - x)}{(1-x^{10})(1-x^9)} \right\\}\] Now, we can see that the term \((1 - x)\) is a common factor in both the numerator and the denominator. Cancel out this term: \[\lim _{x \rightarrow 1} \left\\{\frac{x^9}{(1-x^9)} \right\\}\]

Now we can safely evaluate the limit as x approaches 1. Substitute 1 for x: \[\lim _{x \rightarrow 1} \left\\{\frac{1^9}{1-1^9}\right\\}\] After substituting 1 for x, we get: \[\frac{1}{1-1} = \frac{1}{0}\] However, this is undefined. We have made an error in canceling out the common term in the previous step. Going back to the step, 3: \[\lim _{x \rightarrow 1} \left\\{\frac{1x^9(1 - 10 + 9x)}{(1-x^{10})(1-x^9)} \right\\}\] Let's substitute 1 for x: \[\lim _{x \rightarrow 1} \left\\{\frac{1(1 - 10 + 9)}{(1-1^{10})(1-1^9)} \right\\}\] \[= \frac{1(-9 + 9)}{(1-1)(1-1)}\] \[= \frac{0}{0}\] Since we have 0/0 form, which is an indeterminate form, we need to use L'Hopital's rule. Applying L'Hôpital's Rule: Find the derivative of the numerator and denominator: \[\lim _{x \rightarrow 1} \left\\{\frac{\frac{d}{dx}(1x^9(1 - 10 + 9x))}{\frac{d}{dx}((1-x^{10})(1-x^9))} \right\\}\] Compute the derivatives: \[\lim _{x \rightarrow 1} \left\\{\frac{9x^8(1 - 10 + 9x)}{(-10x^9)(1-x^9) - x^{10}(-9x^8)} \right\\}\] Now, substitute 1 for x: \[\frac{9(1^8)(1 - 10 + 9)}{(-10)(1^9)(1-1^9) - 1^{10}(-9)(1^8)}\] \[= \frac{0}{0}\] We still have the indeterminate form of 0/0, meaning the limit does not exist. In conclusion, the limit does not exist and is not one of the answer choices provided.