Question

Find the numbers at which f is continuous. At which numbers is f discontinuous?

$f\left(x\right)=\frac{x-3}{\ln \left(x\right)}$

Short answer

f is continuous at all positive real numbers $x=1$.

Step-by-step solution

Step 1. Given information 

We have been given a function $f\left(x\right)=\frac{x-3}{\ln \left(x\right)}$.

We have to find the numbers at which f is continuous and at which numbers f is discontinuous.

Step 2. Discontinuous function property

A function f is discontinuous at the number where:

• f is undefined
• the limit does not exist, i.e., its one-sided limits are not equal
• the limit exists, i.e., its one-sided limits equal, but is not equal to the function value

Step 3. Analyze the function

Analyzing the given function,

$f\left(x\right)=\frac{x-3}{\ln \left(x\right)}$

We know that it is a rational function in the form of $R\left(x\right)=\frac{P\left(x\right)}{Q\left(x\right)}$whose domain is

$\left\{x\left|Q\right(x)\ne 0\right\}$

Step 4. Solve for the denominator

Solve for the $Q\left(x\right)$,

$Q\left(x\right)=\ln x$

which implies that x=1,

Also logarithm function takes only positive values,

Thus domain is

$\left\{x|x>0,x\ne 1\right\}$

Thus f is continuous for all real numbers except x=1