# To estimate: The value of lim x → 0 5 x − 4 x 3 x − 2 x using graph and then find the exact value of the same using L’Hospital’s rule.

### 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 48E
To determine

Expert Solution

## Answer to Problem 48E

The exact value of limx05x4x3x2x0.56 .

### Explanation of Solution

Use the online graphing calculator and draw the graph of 5x4x3x2x as shown below in Figure 1.

From Figure 1, it is identified that curve approaches approximately 0.55 as x approaches 0.

Find the exact value with the help of L’Hospital’s rule as follows.

Obtain the value of the function as x approaches 0 .

As x approaches 0 , the numerator is,

5x4x=5040=11=0

As x approaches 0 , the denominator is,

3x2x=3020=11=0 .

Thus, limx05x4x3x2x=00 is in an indeterminate form.

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

limx05x4x3x2x=limx05xln54xln43xln32xln2=50ln540ln430ln320ln2=ln5ln4ln3ln2

Simplify further as follows.

limx05x4x3x2x=ln5ln4ln3ln2=1.611.381.100.69=0.230.410.56

Thus, limx05x4x3x2x0.56 .

Therefore, the exact value of limx05x4x3x2x0.56 .

### Have a homework question?

Subscribe to bartleby learn! Ask subject matter experts 30 homework questions each month. Plus, you’ll have access to millions of step-by-step textbook answers!