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Chapter 10: Q. 92 (page 777)

Use L鈥橦opital鈥檚 rule to prove that every power function 藛 with a positive power dominates the logarithmic functiong(x)=lnx

Short Answer

Expert verified

Every power functions xndominate the logarithmic function lnx.

Step by step solution

01

Step 1. Given information

Exponential growth function and power function are exandxn,n鈭�Z respectively.

02

Step 2. Calculation

Consider,

limx鈫�鈭�xnlnx(鈭�鈭�form);n鈭�Z+

Applying Hopital's rule

=limx鈫�鈭�nxn-11x

Again Applying Hopital's rule,

=limx鈫�鈭�nxn

since n is positive

=鈭�鈭赌x鈭�[0,1]=limx鈫�鈭�xnlnx=鈭�

Which is possible only if xndominates lnx.

Thus, every power functions xndominate the logarithmic function lnx.

Hence, theorem proved.

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Most popular questions from this chapter

In Exercises 36鈥�41 use the given sets of points to find:

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