Question
Upstudy AI Solution
Tutor-Verified Answer
Answer
The limit is
or
.
Solution
Calculate the limit
.
Evaluate the limit by following steps:
Evaluate the limit by following steps:
- step0: Evaluate using L’Hopital’s rule:
- step1: Use the L’Hopital’s rule:
- step2: Find the derivative:
- step3: Use the L’Hopital’s rule:
- step4: Find the derivative:
- step5: Simplify the expression:
- step6: Use the L’Hopital’s rule:
- step7: Find the derivative:
- step8: Rewrite the expression:
- step9: Rewrite the expression:
- step10: Calculate:
The limitis equal to or .
Answered by UpStudy AI and reviewed by a Professional Tutor
Like
error msg


Beyond the Answer
To evaluate the limit
we start by recalling the Taylor series expansion of
around
:
Next, we subtract
from
:
Now, we substitute this expression into our limit:
As
approaches
, the terms
and
go to
. Thus, we can take the limit:
Therefore, the answer is