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# Find the limit as r approaches 0?

((a^r + b^r)/2)^(1/r)

### 3 Answers

- RaffaeleLv 78 years agoFavorite Answer
lim ((a^r + b^r)/2)^(1/r) (as r --> 0)

LN lim = lim LN

lim LN ((a^r + b^r)/2)^(1/r) =

= lim (1/r)LN ((a^r + b^r)/2) (as r --> 0) is a indetermination form 0/0

L'Hopital rule

lim ((a^r LN(a) + b^r LN(b))/(a^r + b^r)) (as r --> 0)

= LN(a)/2 + LN(b)/2 = LN √(ab)

as

LN lim ((a^r + b^r)/2)^(1/r) = LN √(ab)

lim ((a^r + b^r)/2)^(1/r) = √(ab)

- Anonymous6 years ago
43fdas

- Iggy RockoLv 78 years ago
L = lim ((a^r + b^r)/2)^(1/r)

......r-> 0

lnL = lim ln(((a^r + b^r)/2)^(1/r))

........r-> 0

lnL = lim (1/r)ln((a^r + b^r)/2)

........r-> 0

lnL = lim ln((a^r + b^r)/2) / r...now we can use L'Hopital's rule

........r-> 0

lnL = lim ((lna(a^r) + lnb(b^r))/2) / 1 = (lna + lnb)/2

........r-> 0

e^(lnL) = e^((lna + lnb)/2)

L = e^(lna/2)e^(lnb/2)

L = (e^lna - e^(1/2))(e^lnb - e^(1/2))

L = (a - e^(1/2))(b - e^(1/2))

L = ab - (a - b)e^(1/2) + e