Question

find the limit. (lim_{c \to 5} \frac{\frac{1}{c} - \frac{1}{5}}{c - 5})

Explanation:

Step1: Simplify the numerator

First, we simplify the numerator \(\frac{1}{c}-\frac{1}{5}\). Find a common denominator, which is \(5c\). So we have \(\frac{5 - c}{5c}\).
Now the limit becomes \(\lim_{c
ightarrow5}\frac{\frac{5 - c}{5c}}{c - 5}\).

Step2: Rewrite the expression

Notice that \(5 - c=-(c - 5)\). So we can rewrite the numerator as \(\frac{-(c - 5)}{5c}\). Then the expression is \(\lim_{c
ightarrow5}\frac{\frac{-(c - 5)}{5c}}{c - 5}\).
Dividing by \(c - 5\) is the same as multiplying by \(\frac{1}{c - 5}\), so we get \(\lim_{c
ightarrow5}\frac{-(c - 5)}{5c(c - 5)}\).

Step3: Cancel out common factors

We can cancel out the \((c - 5)\) terms (since \(c
eq5\) when taking the limit, we can do this cancellation). So we are left with \(\lim_{c
ightarrow5}\frac{-1}{5c}\).

Step4: Evaluate the limit

Now we substitute \(c = 5\) into \(\frac{-1}{5c}\). So we have \(\frac{-1}{5\times5}=-\frac{1}{25}\).

Answer:

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