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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

Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.

lim x ( 1 + a x ) b x

To determine

To evaluate: The value of limx(1+ax)bx.

Explanation

Given:

Let y=limx(1+ax)bx (1)

Calculation:

Take natural logarithm on both sides,

lny=ln(limx(1+ax)bx)=limx(ln(1+ax)bx)=limx(bxln(1+ax))=limxln(1+ax)1bx

Therefore, lny=limxln(1+ax)1bx (2)

Obtain the value of the function as x approaches .

As x approaches , the numerator is,

limxln(1+ax)=ln(1+a)=ln1=0

As x approaches , the denominator is,

limx1bx=1=0

Thus, limxln(1+ax)1bx=00 is in an indeterminate form

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