Finding Limits of Differences When x → ±∞


Find the limits in Exercises 84–90. (Hint: Try multiplying and dividing by the conjugate.)


lim x → −∞ (2x + √(4x² + 3x − 2))

Verified step by step guidance

1. Identify the expression for which you need to find the limit: \( \lim_{x \to -\infty} (2x + \sqrt{4x^2 + 3x - 2}) \).
2. To simplify the expression, multiply and divide by the conjugate: \( \frac{(2x + \sqrt{4x^2 + 3x - 2})(2x - \sqrt{4x^2 + 3x - 2})}{2x - \sqrt{4x^2 + 3x - 2}} \).
3. The numerator becomes a difference of squares: \((2x)^2 - (\sqrt{4x^2 + 3x - 2})^2 = 4x^2 - (4x^2 + 3x - 2)\). Simplify this to \(-3x + 2\).
4. Now, the expression is \( \frac{-3x + 2}{2x - \sqrt{4x^2 + 3x - 2}} \). Divide every term by \(x\) to simplify: \( \frac{-3 + \frac{2}{x}}{2 - \sqrt{4 + \frac{3}{x} - \frac{2}{x^2}}} \).
5. Evaluate the limit as \(x \to -\infty\). The terms \(\frac{2}{x}\), \(\frac{3}{x}\), and \(\frac{2}{x^2}\) approach zero, simplifying the expression to \( \frac{-3}{2 - 2} \).

Key Concepts

Limits at Infinity

Limits at infinity involve finding the behavior of a function as the variable approaches positive or negative infinity. This concept is crucial for understanding how functions behave asymptotically, often simplifying expressions to determine dominant terms that dictate the limit. In this problem, we analyze the expression as x approaches negative infinity.

Conjugate Multiplication

Multiplying and dividing by the conjugate is a technique used to simplify expressions, especially those involving square roots. The conjugate of a binomial expression like a + b√c is a - b√c, and using it can help eliminate radicals or simplify complex fractions. This method is suggested in the hint to tackle the given limit problem.

Dominant Term Analysis

Dominant term analysis involves identifying the term in an expression that grows fastest as the variable approaches infinity. This helps in approximating the limit by focusing on the most significant contributors to the function's behavior. In the given problem, analyzing the dominant terms in the expression 2x + √(4x² + 3x − 2) is essential for finding the limit as x approaches negative infinity.