Math Problem Statement
Solution
The problem in the image is the following limit expression:
To solve this limit as , we use asymptotic analysis and simplifying techniques. The key idea is to factor out terms that dominate as and deal with the square roots.
Step-by-step solution:
1. Focus on the numerator:
Factor out from the square root: As , the terms and become negligible, so: Thus, the expression becomes: So, the numerator approaches as .
2. Focus on the denominator:
Again, factor out from the square root: As , becomes negligible, so: Thus, the denominator becomes: So, the denominator approaches as .
3. Combine the results:
Now, the limit can be simplified:
Final answer:
Would you like more details on any part of the process?
Related questions:
- How does the technique of factoring out dominant terms help in limits involving square roots?
- What are the most common ways to simplify expressions involving large ?
- Why do terms like and become negligible as ?
- Can this limit be solved using L'Hôpital's Rule instead? How?
- What are the potential pitfalls of not simplifying correctly in limits with square roots?
Tip:
When dealing with limits as , always consider the highest-order terms first, as lower-order terms often become negligible.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Limits
Asymptotic Analysis
Square Roots Simplification
Formulas
lim(x -> +∞) of f(x) / g(x) when dealing with dominant terms
sqrt(a^2 + b) ≈ a(1 + b/(2a^2)) for large a
Theorems
Simplification of limits by factoring dominant terms
Asymptotic approximations for square roots
Suitable Grade Level
Undergraduate/Graduate
Related Recommendation
Limit of (sqrt(x^2 + x + 1)/x - (x + 1)/x) as x approaches infinity
Limit of (x sqrt(x+1)(1 - sqrt(2x+3)))/(7 - 6x + 4x^2) as x Approaches Infinity
Limit of sqrt(x^2 + x) - (x^2 - 2x + 1) as x approaches infinity
Limit of (sqrt(x) + sqrt(x)) / sqrt(2x) as x Approaches Infinity
Calculate the Limit as x Approaches Infinity for the Expression √(x²+x) - (x²-2x+1)