Math Problem Statement
Évalue cette limite
Solution
We are tasked with evaluating the limit:
To simplify this expression and find the limit, we can multiply by the conjugate of the expression. This will help eliminate the square roots:
Multiplying both the numerator and denominator by the conjugate, we get:
Simplifying the terms in the numerator:
Now let's analyze this expression as :
- The numerator approaches , since for large .
- The denominator approaches , because and .
Thus, the expression becomes:
Factor out of the denominator:
As , , so the entire expression tends to . Therefore, the limit is:
Let me know if you want further details or have any questions.
Here are 5 questions to deepen your understanding:
- Why do we use the conjugate method when dealing with square roots in limits?
- How does factoring out simplify the expression?
- What happens to the terms inside the square root as ?
- What are some other methods to evaluate limits with square roots?
- Can the limit of a square root expression approach a finite number when ?
Tip: Multiplying by the conjugate is a powerful tool for simplifying expressions involving square roots, especially when approaching limits at infinity.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Calculus
Limits
Square Roots
Formulas
sqrt(x + 1) - sqrt(2x)
Theorems
Limit properties
Suitable Grade Level
University Level
Related Recommendation
Limit of sqrt(5x + 2) - sqrt(3x + 1) as x approaches infinity
Limit of (sqrt(1+x) - sqrt(1-x)) / x as x approaches 0
Limit of Square Root Expression at Infinity Using Conjugates
Solve Limit Problem Involving Square Roots: sqrt(x+2) - sqrt(2) / x(1+x)
Evaluate the Limit Involving Absolute Value and Square Roots