Question

1. $lim_{x

ightarrow3}\frac{\frac{1}{x}-\frac{1}{3}}{x - 3}$

Explanation:

Step1: Combine fractions in numerator

First, combine $\frac{1}{x}-\frac{1}{3}$: $\frac{1}{x}-\frac{1}{3}=\frac{3 - x}{3x}$.
So the original limit becomes $\lim_{x
ightarrow3}\frac{\frac{3 - x}{3x}}{x - 3}$.

Step2: Simplify the complex - fraction

$\frac{\frac{3 - x}{3x}}{x - 3}=\frac{3 - x}{3x(x - 3)}$. Since $3 - x=-(x - 3)$, we have $\frac{3 - x}{3x(x - 3)}=-\frac{1}{3x}$.

Step3: Evaluate the limit

Now, find $\lim_{x
ightarrow3}-\frac{1}{3x}$. Substitute $x = 3$ into $-\frac{1}{3x}$, we get $-\frac{1}{3\times3}=-\frac{1}{9}$.

Answer:

$-\frac{1}{9}$