Other approaches to evaluate lim_(h => 0) (4^(x+h)+4^(x-h)-4^(x+(1)/(2)))/(h^2)

Trevor Rush 2022-08-11 Answered
Other approaches to evaluate lim h 0 4 x + h + 4 x h 4 x + 1 2 h 2
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Answers (2)

Tristian Clayton
Answered 2022-08-12 Author has 12 answers
A technique that requires no L'Hopital, Taylor series or anything more complicated that the definition of limit, knowledge of the natural logarithm (e.g. from 1 2 d t / t = ln 2 ) and continuity of x x 2
4 x lim h 0 4 h + 4 h 2 h 2 = 4 x lim h 0 ( 4 h 1 ) 2 h 2 4 h = 4 x ( lim h 0 4 h 1 h 2 h ) 2 = 4 x 4 ( lim h 0 2 h 2 h 2 h ) 2 = 4 x + 1 ( ln 2 ) 2
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Elisabeth Wiley
Answered 2022-08-13 Author has 1 answers
A possible way is using the Taylor expansion
2 cosh t = e t + e t = 2 + t 2 + o ( t 2 )
Hence,
4 x + h + 4 x h 4 x + 1 2 h 2 = 4 x e h ln 4 + e h ln 4 2 h 2 = 4 x 2 + h 2 ln 2 4 + o ( h 2 ) 2 h 2 = 4 x ln 2 4 + o ( 1 ) h 0 4 x ln 2 4
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