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 xx \to -\infty. Here's how I interpret your expression:

limxx242x+5\lim_{{x \to -\infty}} \sqrt{\frac{x^2 - 4}{2x + 5}}

Let me solve this step by step:

Step 1: Simplifying the expression inside the square root

We have:

x242x+5\frac{x^2 - 4}{2x + 5}

As xx \to -\infty, the term x2x^2 dominates over the constant 4-4 in the numerator. So:

x24x2x^2 - 4 \approx x^2

Step 2: Handling the denominator

In the denominator, as xx \to -\infty, the linear term 2x2x dominates over the constant 55. So:

2x+52x2x + 5 \approx 2x

Step 3: Simplifying the fraction

Now, we approximate the fraction:

x22x=x2\frac{x^2}{2x} = \frac{x}{2}

Step 4: Applying the square root

We now take the square root of the simplified expression:

x2=x2\sqrt{\frac{x}{2}} = \frac{\sqrt{x}}{\sqrt{2}}

However, since xx \to -\infty, we have to consider the fact that x\sqrt{x} is undefined for negative numbers. But, we can express x2=x\sqrt{x^2} = |x|, and since xx \to -\infty, x=x|x| = -x. Therefore:

x22x=x2\sqrt{\frac{x^2}{2x}} = \frac{-x}{\sqrt{2}}

Step 5: Final answer

Thus, as xx \to -\infty, the limit becomes:

limxx242x+5=x2+\lim_{{x \to -\infty}} \sqrt{\frac{x^2 - 4}{2x + 5}} = \frac{-x}{\sqrt{2}} \to +\infty

So, the limit is ++\infty.

Would you like further clarification or details on any step?

Here are some related questions:

  1. How does one handle square roots in limits approaching negative infinity?
  2. What role does dominance of terms play in simplifying limits?
  3. Can limits approach infinity in functions with square roots?
  4. What happens if the expression inside the square root is negative?
  5. How does the sign of xx 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