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Please find the limit of the question lim x->0 (((e^x+(e^-x)-2)/x^2)^(1/x^2))

Please find the limit of the question
lim x->0 (((e^x+(e^-x)-2)/x^2)^(1/x^2))

Grade:12

1 Answers

Samyak Jain
333 Points
4 years ago
 lim  { (ex + e–x – 2)/x2 }^(1/x2). It is of 1\infty.
x\rightarrow0
So, its solution is  e^(lim {(ex + e–x – 2)/x2  – 1}.(1/x2))  =  e^(lim {(ex + e–x – 2 – x2) / x2}.(1/x2))
                              x\rightarrow0
         =  e^(lim { (ex + e–x – 2 – x2) / x4 })
                x\rightarrow0
Use expansion of ex here, i.e.,
ex = 1 + x + x2/2! + x3/3! + x4/4! + …   and   e–x = 1 – x + x2/2! – x3/3! + x4/4! + …
Subtitute in the limit.
e^(lim { (ex + e–x – 2 – x2) / x4 }) =
   x\rightarrow0
e^(lim { (1 + x + x2/2! + x3/3! + x4/4! + … + 1 – x + x2/2! – x3/3! + x4/4! + … – 2 – x2) / x4 })
   x\rightarrow0
 =  e^(lim { (2.x4/4! + ...) / x4
        x\rightarrow0
 =  e^(1/12).  As x tends to zero higher powers of of x will also tend to zero, so they aren’t written here.

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