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Chapter 14: Q. 72 (page 890)

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

f(x)=lnxx-3

Short Answer

Expert verified

f is continuous at all positive real numbers except x=3.

Step by step solution

01

Step 1. Given information 

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

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)=lnxx-3

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

x|Q(x)≠0

04

Step 4. Solve for the denominator

Solve for the Q(x),

Q(x)=x-3

Which implies that,

x≠3

And all logarithm functions take only positive values.

Thus f is continuous at all positive real numbers except x=3

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Most popular questions from this chapter

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