Math Problem Statement
lim x– $$$$ \left(-\infty\right)\text{ (}\sqrt{\frac{x^2-4)}{2x+5}}
Solution
It seems like there is a formatting issue with the mathematical expression. Based on what you provided, it appears you're asking to compute the limit of a function as . Here's how I interpret your expression:
Let me solve this step by step:
Step 1: Simplifying the expression inside the square root
We have:
As , the term dominates over the constant in the numerator. So:
Step 2: Handling the denominator
In the denominator, as , the linear term dominates over the constant . So:
Step 3: Simplifying the fraction
Now, we approximate the fraction:
Step 4: Applying the square root
We now take the square root of the simplified expression:
However, since , we have to consider the fact that is undefined for negative numbers. But, we can express , and since , . Therefore:
Step 5: Final answer
Thus, as , the limit becomes:
So, the limit is .
Would you like further clarification or details on any step?
Here are some related questions:
- How does one handle square roots in limits approaching negative infinity?
- What role does dominance of terms play in simplifying limits?
- Can limits approach infinity in functions with square roots?
- What happens if the expression inside the square root is negative?
- How does the sign of affect the final result?
Tip: Always check whether the terms inside square roots are positive or negative, especially when approaching negative infinity.
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Math Problem Analysis
Mathematical Concepts
Limits
Infinity
Square Roots
Dominance of Terms
Formulas
lim_{x → -∞} √((x^2 - 4)/(2x + 5))
lim_{x → -∞} (x^2 - 4) ≈ x^2
lim_{x → -∞} (2x + 5) ≈ 2x
Theorems
Limit Properties
Square Root Properties for Large Values
Suitable Grade Level
Undergraduate Calculus or Advanced High School
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