Question

The value of `lim_{x→0}{(a^x + b^x + c^x + d^x)/4}^{1/x}` is ______ 

Options

• abcd

• (abcd)^1/4

• `1/4`abcd

• `1/4`

Answer 1

The value of `lim_{x→0}{(a^x + b^x + c^x + d^x)/4}^{1/x}` is (abcd)^1/4

Explanation:

`lim_{x→0}{(a^x + b^x + c^x + d^x)/4}^{1/x}`

= `lim_{x→0}{1 + (a^x + b^x + c^x + d^x - 4)/4}^{1/x}`

= `lim_{x→0}{1 + ((a^x - 1) + (b^x - 1) + (c^x - 1) + (d^x - 1))/4}^{1/x}`

= `e^{lim_{x→0}((a^x - 1)/(4x) + (b^x - 1)/(4x) + (c^x - 1)/(4x) + (d^x - 1)/(4x)))`

= `e^{log("abcd")^{1/4}} = ("abcd")^{1/4}`