Math Problem Statement
Solution
To evaluate the limit
using the Maclaurin series for , let’s proceed step by step.
1. The Maclaurin Series for
The Maclaurin series expansion of is:
Now, replace with to expand :
2. Simplify the Numerator :
From the series for , subtract and :
Simplify:
3. Divide by :
We now divide the entire expression by :
Separate the terms:
Simplify each term:
4. Take the Limit as :
As , the higher-order terms involving vanish:
Final Answer:
Would you like a detailed explanation of any specific step? Let me know!
Related Questions:
- How do you derive the Maclaurin series for ?
- What is the general form of the Maclaurin series for any function?
- How can the limit of indeterminate forms involving series be solved?
- What are other applications of the Maclaurin series?
- How can higher-order terms affect the convergence of series approximations?
Tip: Always expand the function carefully to include sufficient terms to evaluate limits accurately.
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Math Problem Analysis
Mathematical Concepts
Limits
Maclaurin Series
Calculus
Formulas
Maclaurin series for e^x: \( e^x = \sum_{n=0}^\infty \frac{x^n}{n!} \)
Limit simplification using series expansion
Theorems
Maclaurin Series Expansion
Limit Theorem
Suitable Grade Level
College Level (Calculus I/II)
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