Math Problem Statement
lim((x+ln(sqrt(x^2+1)-x))/(x^3)) when x goes to 0
Solution
To solve the limit
we can break it down and simplify it.
Step 1: Simplify the Logarithmic Term
First, consider the term inside the logarithm:
Using the identity to simplify:
Thus,
Step 2: Expand Around
Using the Taylor series for as :
Thus,
Using the Taylor expansion for when :
Thus,
Step 3: Substitute and Simplify the Limit Expression
Now we rewrite the original limit expression using this approximation:
This simplifies to:
Therefore, the answer is:
Would you like more details or have any questions?
Related Questions:
- How can we apply L'Hôpital's Rule to solve similar indeterminate forms?
- What are some alternative methods for simplifying limits with logarithmic terms?
- How does the Taylor series expansion help in evaluating limits?
- Can this approach be applied to higher-order terms in the Taylor series?
- What other indeterminate forms commonly appear in limit problems?
Tip:
When dealing with limits that involve logarithms or square roots, simplifying by rationalizing or using series expansion often makes complex expressions more manageable.
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Math Problem Analysis
Mathematical Concepts
Calculus
Limits
Logarithmic functions
Taylor Series Expansion
Formulas
Limit definition for indeterminate forms
Taylor series expansion for logarithmic and square root functions around x = 0
Theorems
Taylor Series Expansion
Logarithmic Properties
Limit Laws
Suitable Grade Level
Undergraduate Calculus
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