Finding the value of constant k as x approaches infinity?

Question

#lim_(x->oo) (1+1/(kx))^x = sqrt(e)#

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Answer 1 · 2018-01-21T17:09:04

#k=2#.

A common identity in mathematics is

#lim_(x->oo)(1+n/x)^x=e^n#

If we take #n=1/2#, the equation becomes

#lim_(x->oo)(1+(1//2)/x)^x=e^(1//2)#

which can be rewritten as

#lim_(x->oo)(1+1/(2x))^x=sqrt e#

Thus, the value for #k# we seek is #2#.