Answer: The required value of the limit is 18.
Step-by-step explanation: We are given to find the limit of the following function algebraically :
limit as x approaches nine of quantity x squared minus eighty one divided by quantity x minus nine.
We will be using the following factorization formula :
[tex]a^2-b^2=(a+b)(a-b).[/tex]
The limit can be calculated as follows :
[tex]\ell\\\\\\=\lim_{x\rightarrow 81}\dfrac{x^2-81}{x-9}\\\\\\=\lim_{x\rightarrow 9}\dfrac{x^2-9^2}{x-9}\\\\\\=\lim_{x\rightarrow 9}\dfrac{(x+9)(x-9)}{(x-9)}\\\\=\lim_{x\rightarrow 9}(x+9)~~~~~~~~~~~~~~~~[\textup{since }x\rightarrow 9,~so~x\neq 9]\\\\=9+9\\\\=18.[/tex]
Thus, the required value of the limit is 18.
