# To illustrate: L’Hospitals rule by ploting f ( x ) g ( x ) and f ′ ( x ) g ′ ( x ) near x = 0 to check whether the limits are the same and calculate the exact value of the limit. ### 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 4.5, Problem 49E
To determine

## To illustrate: L’Hospitals rule by ploting f(x)g(x) and f′(x)g′(x) near x=0 to check whether the limits are the same and calculate the exact value of the limit.

Expert Solution

The exact value of limx0ex1x3+4x=0.25 .

### Explanation of Solution

Given:

The functions are f(x)=ex1 and g(x)=x3+4x .

Calculation:

The value of f(x)g(x)=ex1x3+4x .

The value of, f(x)g(x)=ex3x2+4 .

Use the online graphing calculator and draw the graph of f(x)g(x)=ex1x3+4x and f(x)g(x)=ex3x2+4 on the same plane as shown below in Figure 1. From Figure 1, it is identified that both curves approaches 0.25 as x approaches 0. Thus, f(x)g(x) and f(x)g(x) both approaches the same value.

Find the exact limit as follows.

Obtain the value of the function as x approaches 0 .

As x approaches 0, the numerator is,

ex1=e01=11=0

As x approaches 0, the denominator is,

x34x=(0)34(0)=00=0

Thus, limx0ex1x3+4x=00 is in an indeterminate form.

Therefore, apply L’Hospital’s Rule and obtain the limit.

limx0ex1x3+4x=limxex3x2+4=e03(02)+4=14=0.25

Thus, limx0ex1x3+4x=0.25 .

Therefore, the exact value of limx0ex1x3+4x=0.25 .

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