Final answer:
To find the limit as x approaches 81 of the expression (√81 - 9)/(x - 81), we recognize it's an indeterminate form 0/0. By applying L'Hopital's Rule, we differentiate the numerator and the denominator, leading to the limit being zero.
Step-by-step explanation:
To find the lim (x → 81) (√81 - 9)/(x - 81), we first need to simplify the expression where possible. We notice that √81 is equal to 9, so our numerator is 9 - 9 which simplifies to zero. However, when x is 81, the denominator is also zero, leading us to a 0/0 indeterminate form. This suggests we should use algebraic manipulation or apply L'Hopital's Rule, which states that if the limit results in an indeterminate form of 0/0, we can take the derivative of the numerator and the derivative of the denominator and then take the limit.
Since the derivatives of 9 and 81 are 0, and the derivative of x with respect to x is 1, applying L'Hopital's Rule gives us:
lim (x → 81) (0)/(1) = 0
Therefore, the limit of the expression as x approaches 81 is 0.