Question

How do we know when to not use L’hopital’s rule?

Answer 1

Hint: The first thing to try and do is to actually understand after we should use L'Hôpital's Rule. L'Hôpital's Rule may be a brilliant trick for addressing limits of an indeterminate form.
An indeterminate form is when the limit seems to approach a deeply weird answer.

Complete step-by-step solution:
L'Hôpital's Rule says:
If we've got an indeterminate form for our answer to the limit, then we'll take the derivative of the numerator and of the denominator separately so as to search out the limit.
we can repeat this process if we still get an indeterminate form. An indeterminate form is when the limit seems to approach a deeply weird answer.

We must stop as soon as we now not get an indeterminate form by allowing the limit to be reached.
So for my example, we could have used L'Hôpital's Rule:
$\Rightarrow \mathop {\lim }\limits_{\delta x \to 2} \dfrac{{{x^2} - 4}}{{{x^2} - x - 2}}$
Now, take the derivative of the numerator and of the denominator separately so as to search out the limit.
Therefore, we will get,
$\Rightarrow \mathop {\lim }\limits_{\delta x \to 2} \dfrac{{2x}}{{2x - 1}}$

Now, since I've taken the derivative of the numerator and therefore the denominator separately.
Now, substitute $x = 2$, we will get,
$ \Rightarrow \dfrac{4}{3}$
If I had found the solution to still in indeterminate form, then I might need to continue using L'Hôpital's Rule.

Note: If I had found the solution to still in indeterminate form, then I might need to continue using L'Hôpital's Rule. But as soon as I get a determinate form , I have to stop. Because when the solution is not any longer an indeterminate form, L'Hôpital's Rule does not apply.