Finding Limits of Differences When x → ±∞


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


lim x → ∞ (√(9x² − x) − 3x)

Verified step by step guidance

1. Identify the expression whose limit you need to find: \( \lim_{x \to \infty} (\sqrt{9x^2 - x} - 3x) \).
2. Recognize that the expression involves a square root and a linear term, suggesting the use of the conjugate to simplify.
3. Multiply and divide the expression by the conjugate: \( \frac{(\sqrt{9x^2 - x} - 3x)(\sqrt{9x^2 - x} + 3x)}{\sqrt{9x^2 - x} + 3x} \).
4. Simplify the numerator using the difference of squares: \((\sqrt{9x^2 - x})^2 - (3x)^2 = 9x^2 - x - 9x^2 = -x\).
5. Rewrite the expression as \( \frac{-x}{\sqrt{9x^2 - x} + 3x} \) and analyze the behavior of the numerator and denominator as \( x \to \infty \) to find the limit.

Key Concepts

Limits at Infinity

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

Conjugate Method

The conjugate method is a technique used to simplify expressions, especially those involving square roots. By multiplying and dividing by the conjugate, we can eliminate radicals and simplify the expression, making it easier to evaluate limits. This method is particularly useful in rationalizing differences involving square roots, as seen in the given problem.

Dominant Terms

Dominant terms are the parts of an expression that have the most significant impact on its value as the variable approaches infinity. Identifying these terms helps simplify the limit calculation by focusing on the highest degree terms, which dictate the behavior of the function. In this problem, recognizing that 9x² is the dominant term in the square root expression is key to finding the limit.