Question

Can anyone help me understand how to evaluate the limit of this complex fraction?

lim (x-->3)


`(1/x-1/3)/(x-3)`

Answer 1

-1/9

Explanation:

`\lim_(x \to 3) (1/x-1/3)/(x-3)`

For simplicity, let's multiply top and bottom by 3x:

`\lim_(x \to 3) (3-x)/(3x(x-3))`

Factor out a -1:

`\lim_(x \to 3) (-(x-3))/(3x(x-3))`

Divide top and bottom by x−3:

`\lim_(x \to 3) (-1)/(3x)`

Evaluate the limit:

`(-1)/(3(3))\\-(1)/(9)`

It's important to note that the function doesn't exist at x = 3. As x approaches 3, the function approaches -1/9.