# To estimate: The value of lim x → ∞ ( 1 + 2 x ) 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 47E
To determine

## To estimate: The value of limx→∞(1+2x)x using graph and then find the exact value of the same using L’Hospital’s rule.

Expert Solution

The exact value of limx(1+2x)x7.39 .

### Explanation of Solution

Use online graphing calculator and draw the graph of (1+2x)x as shown below in Figure 1.

From Figure 1 it is identified that curve approaches approximately 7 as x approaches ∞.

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

Let, y=limx(1+2x)x . (1)

Take natural logarithm on both sides,

lny=ln(limx(1+2x)x)=limx(ln(1+2x)x)=limx(xln(1+2x))=limxln(1+2x)1x

Therefore, lny=limxln(1+2x)1x . (2)

Obtain the value of the function as x approaches .

As x approaches , the numerator is,

ln(1+2x)=ln(1+2)=ln(1+0)=ln(1)=0

As x approaches , the denominator is,

1x=1=0 .

Thus, limxln(1+2x)1x=00 is in an indeterminate form.

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

limxln(1+2x)1x=limx11+2x(2x2)1x2=limx21+2x1x2x2=limx21+2x=2

Thus, limx1+ln(1+2x)1x=2 .

So, the equation (2) becomes,

lny=2y=e2=(2.72)27.39

Thus, the exact value of limx(1+2x)x7.39 .

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