Question

question
evaluate the limit: (limlimits_{x \to 9} \frac{x - 9}{\frac{6}{5} - \frac{x - 3}{x - 4}})

answer attempt 1 out of 2

dne
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Explanation:

Step1: Simplify the denominator

First, simplify the denominator \(\frac{6}{5}-\frac{x - 3}{x - 4}\). Find a common denominator, which is \(5(x - 4)\).
\[

$$\begin{align*} \frac{6}{5}-\frac{x - 3}{x - 4}&=\frac{6(x - 4)-5(x - 3)}{5(x - 4)}\\ &=\frac{6x-24 - 5x + 15}{5(x - 4)}\\ &=\frac{x-9}{5(x - 4)} \end{align*}$$

\]

Step2: Substitute the simplified denominator into the original limit

The original limit \(\lim_{x
ightarrow9}\frac{x - 9}{\frac{6}{5}-\frac{x - 3}{x - 4}}\) becomes \(\lim_{x
ightarrow9}\frac{x - 9}{\frac{x - 9}{5(x - 4)}}\) (since the denominator simplifies to \(\frac{x - 9}{5(x - 4)}\)).

Step3: Cancel out the common factor

Cancel out the common factor \(x - 9\) (assuming \(x
eq9\), which is valid for the limit as \(x
ightarrow9\) but \(x
eq9\) in the neighborhood). So we get \(\lim_{x
ightarrow9}5(x - 4)\).

Step4: Evaluate the limit

Now, substitute \(x = 9\) into \(5(x - 4)\): \(5\times(9 - 4)=5\times5 = 25\).

Answer:

\(25\)