# Whether the statement “If f has domain [ 0 , ∞ ] and has no horizontal asymptote, then lim x → ∞ f ( x ) = ∞ or lim x → ∞ f ( x ) = − ∞ ” is true or false.

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter 2, Problem 10RQ
To determine

## To find: Whether the statement “If f has domain [0,∞] and has no horizontal asymptote, then limx→∞f(x)=∞ or limx→∞f(x)=−∞ ” is true or false.

Expert Solution

The statement “If f has domain [0,] and has no horizontal asymptote, then limxf(x)= or limxf(x)= ” is true.

### Explanation of Solution

Given information:

The given statement is “If f has domain [0,] and has no horizontal asymptote, then limxf(x)= or limxf(x)= ”.

Calculation:

Consider the function f with domain [0,] and has no horizontal asymptote.

A horizontal asymptote is a horizontal tangent line to the curve at infinity. The line y=L is called a horizontal asymptote of the curve y=f(x) if either limxf(x)=L or limxf(x)=L .

The domain of the f is [0,] . So x varies from 0 to . Since the function does not have the horizontal asymptote, the value of the limit of the function as x is not finite i.e.

limxf(x)=limxf(x)=

Hence, if the function f has domain [0,] and has no horizontal asymptote, then limxf(x)==

Therefore, the statement “If f has domain [0,] and has no horizontal asymptote, then limxf(x)= or limxf(x)= ” is true.

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