Question

evaluate the limit, if it exists. (if an answer does not exist, enter dne.)
\\(lim_{t
ightarrow0}left(\frac{9}{tsqrt{1 + t}}-\frac{9}{t}
ight)\\)

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 \(\lim_{t
ightarrow0}\frac{-81}{\sqrt{1 + t}(9 + 9\sqrt{1 + t})}\). 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:

\(-\frac{9}{2}\)