Question #1488e
1 Answer
The limit does not exist.
Explanation:
We can't simply plug in
However, note that at
Instead, it will be
Let's approach
Let
#x = 2 + a# , where#a# is an extremely small number. Then the negative limit is:
#lim_(a->0^-)((2+a)^(1/3)-2)/a = "negative value"/"negative value approaching 0" = "positive value"/"positive value approaching 0" = oo# Similarly, the positive limit is:
#lim_(a->0^+)((2+a)^(1/3)-2)/a = "negative value"/"positive value approaching 0" = -oo#
Since the negative limit is
To show this limit, here is the graph of the function
graph{(x^(1/3)-2)/(x-2) [-3.184, 6.816, -2.32, 2.68]}
Final Answer