Prove that if limx→∞f(x)=L and L≠0 then limx→∞1f(x)=1L.

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Lesson Parent: 
Function Limits IX: Horizontal Asymptotes And The Limit Of f(x) As x Tends To Infinity
Problem: 

Prove that if
limx→∞f(x)=L
and L≠0 then
limx→∞1f(x)=1L.

Answer: 

It is true that limx→∞1f(x)=1L.

SAMPLE SOLUTION

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Solution: 

Since

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which means that

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and therefore

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Finally, choose Y=Y2 such that <Sign in to see all the formulas>. With <Sign in to see all the formulas> and x such that <Sign in to see all the formulas>, then

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