Question

Lim x+1/(
X->1 sqrt(x^3+8)-3)

Answer 1

I’m not sure whether you mean x + [1/(sqrt(x^3 + 8) – 3)] or
if you mean (x + 1)/(sqrt(x^3 + 8) – 3).

Neither of them has a limit.
This is because the left and right hand limits aren't the same.

I’ll show that (x + 1)/(sqrt(x^3 + 8) – 3) has no limit as x goes to 1.
If you meant the first expression, the reasoning is the same.

Imagine approaching 1 from the left hand side.
Then sqrt(x^3 +8) is always a little less than 3.
So sqrt(x^3 +8) – 3 is always negative as we
let x approach 1 from the left side.

But notice that sqrt(x^3 +8) – 3 gets really small as x goes to 1.
So that means 1/[sqrt(x^3 +8) – 3] goes to negative infinity as
x goes to 1 from the left side.

Now imagine approaching 1 from the right hand side.
Then sqrt(x^3 +8) is always a little bigger than 3.
So sqrt(x^3 +8) – 3 is always positive as we
let x approach 1 from the right side.

But notice that sqrt(x^3 +8) – 3 gets really small as x goes to 1.
So that means 1/[sqrt(x^3 +8) – 3] goes to positive infinity as
x goes to 1 from the left side.

We also know that lim (x + 1) as x goes to 1 is just 2.

So our expression,
numerator is bounded while the denominator
gets infinitely small.

So lim (x + 1)/(sqrt(x^3 + 8) – 3) equals
- infinity as x approaches from the left.
And lim (x + 1)/(sqrt(x^3 + 8) – 3) equals
+ infinity as x approaches from the right.

So there is no limit since the left and right limits aren’t equal.