Question

question 9 evaluate the limit \\( \lim_{x \to 5} \frac{\frac{1}{x + 4} - \frac{1}{9}}{x - 5} \\)

Explanation:

Step1: Combine numerator fractions

First, find a common denominator for the fractions in the numerator, which is $9(x+4)$.
$$\frac{1}{x+4} - \frac{1}{9} = \frac{9 - (x+4)}{9(x+4)} = \frac{9 - x - 4}{9(x+4)} = \frac{5 - x}{9(x+4)}$$

Step2: Rewrite the original limit

Substitute the combined fraction back into the limit, and rewrite the denominator $x-5$ as $-(5-x)$.
$$\lim_{x \to 5} \frac{\frac{5 - x}{9(x+4)}}{x - 5} = \lim_{x \to 5} \frac{5 - x}{9(x+4)(x - 5)} = \lim_{x \to 5} \frac{-(x - 5)}{9(x+4)(x - 5)}$$

Step3: Cancel common factors

Cancel the $(x-5)$ term (valid since $x
eq 5$ when taking the limit).
$$\lim_{x \to 5} \frac{-1}{9(x+4)}$$

Step4: Substitute $x=5$

Plug in $x=5$ into the simplified expression.
$$\frac{-1}{9(5+4)} = \frac{-1}{9 \times 9} = \frac{-1}{81}$$

Answer:

$\frac{-1}{81}$