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Ch. 2 - Limits and Continuity
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 2 - Limits and ContinuityProblem 2.6.87
Chapter 2, Problem 2.6.87

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: \( \lim_{x \to -\infty} (2x + \sqrt{4x^2 + 3x - 2}) \).
To simplify the expression, multiply and divide by the conjugate: \( \frac{(2x + \sqrt{4x^2 + 3x - 2})(2x - \sqrt{4x^2 + 3x - 2})}{2x - \sqrt{4x^2 + 3x - 2}} \).
The numerator becomes a difference of squares: \((2x)^2 - (\sqrt{4x^2 + 3x - 2})^2 = 4x^2 - (4x^2 + 3x - 2)\). Simplify this to \(-3x + 2\).
Now, the expression is \( \frac{-3x + 2}{2x - \sqrt{4x^2 + 3x - 2}} \). Divide every term by \(x\) to simplify: \( \frac{-3 + \frac{2}{x}}{2 - \sqrt{4 + \frac{3}{x} - \frac{2}{x^2}}} \).
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 \( \frac{-3}{2 - 2} \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
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Cases Where Limits Do Not Exist

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.
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Limits of Rational Functions with Radicals

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.
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Derivatives Applied To Acceleration
Related Practice
Textbook Question

Using the Formal Definition


Each of Exercises 31–36 gives a function f(x), a point c, and a positive number ε. Find L = lim x→c f(x). Then find a number δ > 0 such that |f(x)−L| < ε whenever 0 < |x−c| < δ.


f(x) = −3x − 2, c = −1, ε = 0.03

Textbook Question

Horizontal and Vertical Asymptotes

Assume that constants a and b are positive. Find equations for all horizontal and vertical asymptotes for the graph of y = (√ax² + 4) / (x―b) .

Textbook Question

Slope of a Curve at a Point


In Exercises 7–18, use the method in Example 3 to find (a) the slope of the curve at the given point P, and (b) an equation of the tangent line at P.


y=7−x², P(2,3)

Textbook Question

Using the Sandwich Theorem


If 2−x² ≤ g(x) ≤ 2cosx for all x, find limx→0 g(x).

Textbook Question

Limits as x → ∞ or x → −∞


The process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of x. Divide numerator and denominator by the highest power of x in the denominator and proceed from there. Find the limits in Exercises 23–36. Write ∞ or −∞ where appropriate.


lim x→∞ √(x² + 1) / (x + 1)

Textbook Question

Use formal definitions to prove the limit statements in Exercises 93–96.


lim x → 0 (−1 / x²) = −∞