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Use L’Hopital’s rule to prove that every power function ˆ 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,nZ respectively.

02

Step 2. Calculation

Consider,

limxxnlnx(form);nZ+

Applying Hopital's rule

=limxnxn-11x

Again Applying Hopital's rule,

=limxnxn

since n is positive

=x[0,1]=limxxnlnx=

Which is possible only if xndominates lnx.

Thus, every power functions xndominate the logarithmic function lnx.

Hence, theorem proved.

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