De l'Hospital rule applies to undetermined forms like
If we evaluate your limit directly, we have
which is neither of the two forms covered by the theorem.
So, in order to apply it, we need to write the limit as follows: we start with
Using the identity , we can rewrite the function as
Using the rule , we have
Since the exponential function is continuous, we have
In other words, we can focus on the exponent alone to solve the limit. So, we're focusing on
Which we can rewrite as
Now the limit comes in the form 0/0, so we can apply the theorem: we derive both numerator and denominator to get
So, the limit of the exponent is -6, which implies that the whole expression tends to
using l'hospitals rule
lim x --> 0 (1+2x)^(-3/x)...