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

Samyak Jain
333 Points
2 years ago
lim  { (ex + e–x – 2)/x2 }^(1/x2). It is of 1$\dpi{80} \infty$.
x$\dpi{80} \rightarrow$0
So, its solution is  e^(lim {(ex + e–x – 2)/x2  – 1}.(1/x2))  =  e^(lim {(ex + e–x – 2 – x2) / x2}.(1/x2))
x$\dpi{80} \rightarrow$0
=  e^(lim { (ex + e–x – 2 – x2) / x4 })
x$\dpi{80} \rightarrow$0
Use expansion of ex here, i.e.,
ex = 1 + x + x2/2! + x3/3! + x4/4! + …   and   e–x = 1 – x + x2/2! – x3/3! + x4/4! + …
Subtitute in the limit.
e^(lim { (ex + e–x – 2 – x2) / x4 }) =
x$\dpi{80} \rightarrow$0
e^(lim { (1 + x + x2/2! + x3/3! + x4/4! + … + 1 – x + x2/2! – x3/3! + x4/4! + … – 2 – x2) / x4 })
x$\dpi{80} \rightarrow$0
=  e^(lim { (2.x4/4! + ...) / x4
x$\dpi{80} \rightarrow$0
=  e^(1/12).  As x tends to zero higher powers of of x will also tend to zero, so they aren’t written here.