Math Problem Statement
lim x -> infinity(sqrt(1 + x ^ 4) - 1 - x ^ 2)/(x ^ 2) =
A)-1
B0
C-2
D 2
Solution
We are tasked with evaluating the following limit:
Step 1: Simplify the Expression Inside the Limit
First, consider the term inside the square root:
For large values of , the term dominates over , so we can approximate:
Now rewrite the expression:
Step 2: Use Asymptotic Expansion
We can perform an asymptotic expansion for when is large. Let's expand as follows:
So, the approximation becomes:
Step 3: Substitute the Expansion into the Expression
Now substitute this approximation back into the original limit:
Simplify the numerator:
So the expression becomes:
Step 4: Simplify Further
Now break the expression into two terms:
Step 5: Take the Limit as
As , the term approaches 0, and also approaches 0. Therefore, the limit simplifies to:
Thus, the value of the limit is:
The correct answer is B) 0.
Would you like more details or have any questions about the steps? Here are 5 related questions for further practice:
- How do you apply asymptotic expansions in limits?
- What other techniques can be used to evaluate limits involving square roots?
- How do you approach limits at infinity in general?
- Can you apply L'Hopital's Rule to this limit? What would the result be?
- How would you handle a similar limit with a different degree of polynomial inside the square root?
Tip: When working with limits involving square roots, try approximating the square root expression using a binomial expansion or leading-order terms to simplify the analysis.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Limits
Asymptotic Expansion
Infinity
Algebraic Manipulation
Formulas
lim x -> infinity (sqrt(1 + x^4) - 1 - x^2)/(x^2)
Asymptotic expansion: sqrt(1 + x^4) ≈ x^2 + (1/2x^2)
Theorems
Limit Theorem
Asymptotic Approximation
Suitable Grade Level
College-Level Calculus
Related Recommendation
Limit as x Approaches Infinity of sqrt(x^4 + 25) / (1 - x^2)
Limit of (x^2 + 1) / sqrt(x^4 - 2) as x Approaches Infinity
Limit of x(sqrt(4 + 1/x) - 2) as x approaches infinity
Limit of (x - sqrt(x^2 + x + 1)) / (2x + sqrt(4x^2 + x)) as x approaches infinity
Limit of (sqrt(1 + 4x^6))/(2 - x^3) as x approaches negative infinity