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Μάθημα 8: L’Hôpital’s rule: composite exponential functionsL'Hôpital's rule review
L'Hôpital's rule helps us find many limits where direct substitution ends with the indeterminate forms 0/0 or ∞/∞. Review how (and when) it's applied.
What is L'Hôpital's rule?
L'Hôpital's rule helps us evaluate indeterminate limits of the form or .
In other words, it helps us find , where (or, alternatively, where both limits are ).
The rule essentially says that if the limit exists, then the two limits are equal:
Want to learn more about L'Hôpital's rule? Check out this video.
Using L'Hôpital's rule to find limits of quotients
Let's find, for example, .
Substituting into results in the indeterminate form . So let's use L’Hôpital’s rule.
Note that we were only able to use L’Hôpital’s rule because the limit actually exists.
Want to try more problems like this? Check out this exercise.
Using L'Hôpital's rule to find limits of exponents
Let's find, for example, . Substituting into the expression results in the indeterminate form .
To make the expression easier to analyze, let's take its natural log (this is a common trick when dealing with composite exponential functions). In other words, letting , we will find . Once we find it, we will be able to find .
Substituting into results in the indeterminate form , so now it's L’Hôpital’s rule's turn to help us with our quest!
We found that , which means .
Want to try more problems like this? Check out this exercise.
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