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If Limits Do Not Exist

If the limit command cannot compute a limit of a function at a particular point and also cannot prove that the limit is not defined at this point, the command returns an unresolved limit:

limit(gamma(1/x)*cos(sin(1/x)), x = 0)

If limit can prove that the limit is undefined at a particular point, then it returns undefined:

limit(exp(x)*cos(1/x), x = 0)

The function exp(x)*cos(1/x) also does not have one-sided limits at x = 0:

limit(exp(x)*cos(1/x), x = 0, Left);
limit(exp(x)*cos(1/x), x = 0, Right)

The plot shows that as exp(x)*cos(1/x) approaches x = 0, the function oscillates between and :

p1 := plot::Function2d(exp(x)*cos(1/x), x = -PI/4..PI/4):
p2 := plot::Function2d(exp(x), x = -PI/4..PI/4, Color = RGB::Red):
p3 := plot::Function2d(-exp(x), x = -PI/4..PI/4,Color = RGB::Red):
plot(p1, p2, p3)

To get the interval of all possible accumulation points of the function exp(x)*cos(1/x) near the singularity x = 0, use the option Intervals:

limit(exp(x)*cos(1/x), x = 0, Intervals)

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