Chapter 14: Q. 71 (page 890)

Find the numbers at which f is continuous. At which numbers is f discontinuous?
$f (x) = \frac{x - 3}{\ln (x)}$

Short Answer

Expert verified

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

Step by step solution

Step 1. Given information

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

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,

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

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

$\{x | Q (x) \neq 0\}$

Step 4. Solve for the denominator

Solve for the $Q (x)$ ,

$Q (x) = \ln x$

which implies that x=1,

Also logarithm function takes only positive values,

Thus domain is

$\{x | x > 0, x \neq 1\}$

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

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Find the numbers at which f is continuous. At which numbers is f discontinuous?
$f (x) = \frac{x - 3}{\ln (x)}$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Discontinuous function property

Step 3. Analyze the function

Step 4. Solve for the denominator

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Find the numbers at which f is continuous. At which numbers is f discontinuous?f(x)=x-3lnx

Short Answer

Step by step solution

Step 1. Given information

Step 2. Discontinuous function property

Step 3. Analyze the function

Step 4. Solve for the denominator

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Discrete Mathematics

Pure Maths

Decision Maths

Mechanics Maths

Statistics

Study anywhere. Anytime. Across all devices.

Find the numbers at which f is continuous. At which numbers is f discontinuous?
$f (x) = \frac{x - 3}{\ln (x)}$