Assuming x --> 1...
Use the difference of two cubes on (x - 1).
x - 1 = (³√x)³ - 1³ = (³√x - 1)(³√x² + ³√x + 1).
Therefore,
lim(x -->1) (1 - ³√x)/(x - 1)
= lim(x -->1) -(³√x - 1)/[(³√x - 1)(³√x² + ³√x + 1)]
= lim(x -->1) -1 / (³√x² + ³√x + 1)
= -1/(1 + 1 + 1)
= -1/3.
I hope this helps!
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Verified answer
Assuming x --> 1...
Use the difference of two cubes on (x - 1).
x - 1 = (³√x)³ - 1³ = (³√x - 1)(³√x² + ³√x + 1).
Therefore,
lim(x -->1) (1 - ³√x)/(x - 1)
= lim(x -->1) -(³√x - 1)/[(³√x - 1)(³√x² + ³√x + 1)]
= lim(x -->1) -1 / (³√x² + ³√x + 1)
= -1/(1 + 1 + 1)
= -1/3.
I hope this helps!