The limit of the function x^cos(1/x) as x approaches 0 from the positive side is evaluated to be 1. The solution involves taking the natural logarithm of y, leading to the expression lny = cos(1/x) * lnx. By applying the Sandwich Theorem and L'Hôpital's Rule, it is established that both x*cos(1/x) and lnx/x approach 0, confirming that lny = 0 and thus y = 1. The discussion also highlights the behavior of cos(1/x) when negative, indicating that y would tend to infinity, but this does not affect the limit as x approaches 0 from the right.
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No, that doesn't look right at all.Snen said:Homework Statement
Homework Equations
The Attempt at a Solution
let y = lim x->0+ x^cos(1/x)
lny = cos(1/x)*lnx = (x*cos(1/x)) * (lnx/x)
x*cos(1/x) = 0 (sandwich theorem)
lnx/x = 0 (l'hopital)
so lny = 0
and y = 1
Is this correct?
PeroK said:No, that doesn't look right at all.
what happens when ##\cos(1/x)## is negative?
Well then y would tend to infinity but the limit is x-> 0+PeroK said:No, that doesn't look right at all.
what happens when ##\cos(1/x)## is negative?
Snen said:Well then y would tend to infinity but the limit is x-> 0+
tends to infinity I guessRay Vickson said:He is not asking about what happens to x; he is asking about what happens to x^cos(1/x).