BuyFind

Single Variable Calculus: Concepts...

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

Single Variable Calculus: Concepts...

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

Solutions

Chapter 4.5, Problem 48E
To determine

To estimate: The value of limx05x4x3x2x using graph and then find the exact value of the same using L’Hospital’s rule.

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.

Single Variable Calculus: Concepts and Contexts, Enhanced Edition, Chapter 4.5, Problem 48E

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 .

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