Pregunta
What is the limit of the following, using L’Hôpital’s rule?
Ask by Ruiz Fuentes. in the United States
Dec 28,2024
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The limit is 0.
Solución

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To solve the limit
using L’Hôpital’s rule, we first evaluate the limit as
approaches infinity:
Both the numerator
and the denominator
approach infinity, giving us an indeterminate form
.
Applying L’Hôpital’s rule, we differentiate the numerator and the denominator:
- The derivative of the numerator
is . - The derivative of the denominator
is .
Now we can take the limit again:
The new limit is still in the indeterminate form
, so we apply L’Hôpital’s rule a second time:
- The derivative of the numerator
is . - The derivative of the denominator
is still .
Thus, we have:
This limit is still in the
form, so we apply L’Hôpital’s rule again:
- The derivative of the numerator
is . - The derivative of the denominator
remains .
Now we have:
As
approaches infinity,
grows exponentially, so this limit approaches:
Thus, the limit is:
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