Question

question
evaluate the limit: $limlimits_{x \to 1} \frac{\frac{2}{5} - \frac{2}{x + 4}}{2x - 2}$
answer attempt 1 out of 2
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Explanation:

Step1: Simplify the numerator

First, we simplify the numerator \(\frac{2}{5}-\frac{2}{x + 4}\). Find a common denominator, which is \(5(x + 4)\). Then we have:
\[

$$\begin{align*} \frac{2}{5}-\frac{2}{x + 4}&=\frac{2(x + 4)-2\times5}{5(x + 4)}\\ &=\frac{2x+8 - 10}{5(x + 4)}\\ &=\frac{2x-2}{5(x + 4)} \end{align*}$$

\]

Step2: Substitute the simplified numerator into the original limit

The original limit is \(\lim_{x
ightarrow1}\frac{\frac{2}{5}-\frac{2}{x + 4}}{2x - 2}\). Substitute the simplified numerator we got above:
\[
\lim_{x
ightarrow1}\frac{\frac{2x - 2}{5(x + 4)}}{2x - 2}
\]

Step3: Cancel out the common factor

We can cancel out the common factor \(2x - 2\) (note that \(x
ightarrow1\) but \(x
eq1\) when taking the limit, so we can cancel it) :
\[
\lim_{x
ightarrow1}\frac{1}{5(x + 4)}
\]

Step4: Evaluate the limit

Now we can directly substitute \(x = 1\) into the function \(\frac{1}{5(x + 4)}\):
\[
\frac{1}{5(1 + 4)}=\frac{1}{5\times5}=\frac{1}{25}
\]

Answer:

\(\frac{1}{25}\)