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Chapter 14: Q. 71 (page 890)

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

f(x)=x-3lnx

Short Answer

Expert verified

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

Step by step solution

01

Step 1. Given information 

We have been given a function f(x)=x-3lnx.

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

02

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
03

Step 3. Analyze the function

Analyzing the given function,

f(x)=x-3lnx

We know that it is a rational function in the form of R(x)=P(x)Q(x)whose domain is

x|Q(x)鈮�0

04

Step 4. Solve for the denominator

Solve for the Q(x),

Q(x)=lnx

which implies that x=1,

Also logarithm function takes only positive values,

Thus domain is

x|x>0,x鈮�1

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

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