lim x-> 0 (1+x)^(-1/x) = 1/e but how can someone please show me the steps? Solut
ID: 1947164 • Letter: L
Question
lim x-> 0 (1+x)^(-1/x) = 1/e but how can someone please show me the steps?Explanation / Answer
L = lim (1+x)^(1/x), as x->0 e^L = exp{ lim { ln ((1 + x)^{1/x})} } as x-> 0 where I write exp() on the right hand side to mean e^, "the exponential of." I am going to stop writing as x->0 for brevity, but it should be there on your homework, note where the limit operator actually is, it is inside the exponential, and outside the logarithm. e^{ln(L)} = L = exp{ (1/x) ln (1 + x) }, recall ln (a^b) = b ln(a) L = exp{ ( ln (1 + x) ) / x } Noting that ln(1) = 0, as x->0 we have an indeterminate type of 0/0, which permit L'Hopital's rule to be used, differentiate the top and bottom of the fraction independently numerator: d/dx ln (1 + x) = 1 / (1 + x) denominator: d/dx (x) = 1 so we have L = L = exp{ lim ( ln (1 + x) ) / x } as x->0 L = exp{ lim (1 / (1 + x)) } as x->0 L = exp(1) exp(1) is the same as e^1 = e
Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.