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If Limits Do Not Exist

MuPAD® notebooks are not recommended. Use MATLAB® live scripts instead.

MATLAB live scripts support most MuPAD functionality, though there are some differences. For more information, see Convert MuPAD Notebooks to MATLAB Live Scripts.

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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