What is the limit as x approaches 0 of ((a^x+b^x)/2)^{1/x} ?

The limit as x approaches 0 of ((a^x+b^x)/2)^{1/x} is e^{(ln(a)+ln(b))/2}

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limit as x approaches 0 of ((a^x+b^x)/2)^{1/x}

x \to 0 lim ((\frac{a ^{x} + b ^{x}}{2})^{\frac{1}{x}})

e^{\frac{l n ( a ) + l n ( b )}{2}}

Solution steps

x \to 0 lim ((\frac{a ^{x} + b ^{x}}{2})^{\frac{1}{x}})

Apply exponent rules

= x \to 0 lim (e^{\frac{1}{x} l n (\frac{a ^{x} + b ^{x}}{2})})

Apply the Limit Chain Rule:

e^{\frac{l n ( a ) + l n ( b )}{2}}

= e^{\frac{l n ( a ) + l n ( b )}{2}}

What is the limit as x approaches 0 of ((a^x+b^x)/2)^{1/x} ?
The limit as x approaches 0 of ((a^x+b^x)/2)^{1/x} is e^{(ln(a)+ln(b))/2}