Question

Question #9d392

Answer 1

The `limx rarr 0` does not exist as

`lim x rarr 0^+ ((2^-x - 10^x)/2)^(1/x)` yields 0

while

`lim x rarr 0^(-) ((2^-x - 10^x)/2)^(1/x)` yields `oo`

Explanation:

Graphing `((2^-x - 10^x)/2)^(1/x)` we can side as the function approaches 0 from the left it goes to infinity while on the right side it approaches 0. Because the LHL `!=` RHL the limit at x = 0 does not exist.

graph{((2^(-x) - 10^(x))/2)^(1/x) [-10, 10, -5, 5]}

You can also use a tabular method of inserting numbers from the left such as -.00000000000001 and from the right such as .00000000000001 and you'll see they do not approach the same values.