evaluate the limit, if it exists. (if an answer does not exist, enter dne.)\n\\(lim_{t\rightarrow0}left(\frac…

Explanation:

Step1: Combine the fractions

First, find a common - denominator for the two fractions. The common denominator of (t\sqrt{1 + t}) and (t) is (t\sqrt{1 + t}). So, (\frac{9}{t\sqrt{1 + t}}-\frac{9}{t}=\frac{9 - 9\sqrt{1 + t}}{t\sqrt{1 + t}}).

Step2: Rationalize the numerator

Multiply the numerator and denominator by the conjugate of the numerator (9 + 9\sqrt{1 + t}). [ \begin{align*} &\frac{(9 - 9\sqrt{1 + t})(9 + 9\sqrt{1 + t})}{t\sqrt{1 + t}(9 + 9\sqrt{1 + t})}\ =&\frac{81-81(1 + t)}{t\sqrt{1 + t}(9 + 9\sqrt{1 + t})}\ =&\frac{81-81 - 81t}{t\sqrt{1 + t}(9 + 9\sqrt{1 + t})}\ =&\frac{- 81t}{t\sqrt{1 + t}(9 + 9\sqrt{1 + t})} \end{align*} ]

Step3: Simplify the fraction

Cancel out the common factor (t) in the numerator and denominator. We get (\frac{-81}{\sqrt{1 + t}(9 + 9\sqrt{1 + t})}).

Step4: Evaluate the limit

Now, find (\lim_{t\rightarrow0}\frac{-81}{\sqrt{1 + t}(9 + 9\sqrt{1 + t})}). Substitute (t = 0) into the expression: (\frac{-81}{\sqrt{1+0}(9 + 9\sqrt{1+0})}=\frac{-81}{1\times(9 + 9)}=\frac{-81}{18}=-\frac{9}{2}).

Answer:

(-\frac{9}{2})