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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

Illustrate l’Hospital’s Rule by graphing both f(x)/g(x) and f′(x)/g′(x) near x = 0 to see that these ratios have the same limit as x → 0. Also, calculate the exact value of the limit.

f(x) = 2x sin x, g(x) = sec x − 1

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.

Explanation

Given:

The functions are f(x)=2xsinx and g(x)=secx1 .

Calculation:

The value of f(x)g(x)=2xsinxsecx1 .

The value of, f(x)g(x)=2xcosx+2sinxsecxtanx .

Use the online graphing calculator and draw the graphs of f(x)g(x)=2xsinxsecx1 and f(x)g(x)=2xcosx+2sinxsecxtanx on the same plane as shown below in Figure 1.

From Figure 1, it is identified that both curves approach to 4 as x approaches 0. Thus, f(x)g(x) and f(x)g(x) approaches the same limit.

Find the exact limit 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,

2xsinx=20sin0=0

As x approaches 0, the denominator is,

secx1=sec01=11=0 .

Thus, limx02xsinxsecx1=00 is in an indeterminate form.

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

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