Conflicting limit answers using calculator and wolfram alpha

I want to evaluate $\lim\limits_{x\to0} \dfrac{\tan(x) - \sin(x)}{(\sin(x))^3}$,

Calculator says it's 0 when substituted with 0.0000000001.

Wolfram Alpha says it's 1/2.

The Problem Set says the answer is 1/2.

I think I believe Wolfram Alpha more but I've been using the calculator method so I can answer stuff really fast (because it's for a board exam, shouldn't spend too much time deriving) is there a way for me to know?

As a rule of thumb, try express everything in term of either $\sin(x)$ or $\cos(x)$ to see whether there is any obvious cancellation. For this case, we have $$\frac{\tan(x) - \sin(x)}{\sin(x)^3} = \frac{\frac{\sin(x)}{\cos(x)} - \sin(x)}{\sin (x)^3} = \frac{1-\cos(x)}{\cos(x)(1-\cos(x)^2)} = \frac{1}{\cos(x)(1+\cos(x))}$$ you don't need any calculator to know the limit is $\frac12$.

$0.0000000001$ is too small of a number: the calculator got such a small answer for the top that it assumed it was zero (since the values subtracted in the numerator were rounded to the same value). The bottom was non-zero, so there was no division by zero error. Zero divided by anything non-zero is zero.

If you're going to use the calculator method, I would try with a bigger number. I think $10^{-5} = 0.00001$ should be small enough to give you a good answer without causing you to run into this situation.