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

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

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

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

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

Solution steps

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

Simplify

\frac{( a ^{\frac{1}{x}} + b ^{\frac{1}{x}} ) ^{x}}{2 ^{x}} : (\frac{a ^{\frac{1}{x}} + b ^{\frac{1}{x}}}{2})^{x}

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

Apply exponent rules

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

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