Question

Question #1488e

Answer 1

The limit does not exist.

Explanation:

We can't simply plug in `x=2`, because this will give us a denominator of `0`.

However, note that at `x=2`, the term `x^(1/3) - 2` will NOT be `0`.
Instead, it will be `2^(1/3) - 2`, which is less than `0`.

Let's approach `x=2` from both sides:

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 `oo` and the positive limit is `-oo`, the two limits are not the same and therefore we say the limit does not exist.

To show this limit, here is the graph of the function `y = (x^(1/3)-2)/(x-2)`
graph{(x^(1/3)-2)/(x-2) [-3.184, 6.816, -2.32, 2.68]}

Final Answer