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Ch. 2 - Limits and Continuity
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 2 - Limits and ContinuityProblema 2.6.87
Capitolo 2, Problema 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))

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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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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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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) .

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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)

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Using the Sandwich Theorem


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

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Limits with trigonometric functions


Find the limits in Exercises 43–50.


lim x→0 tan x

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Using limθ→0 sin θ / θ = 1


Find the limits in Exercises 23–46.


limθ→0 sin θ / sin 2θ

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Finding Deltas Algebraically


Each of Exercises 15–30 gives a function f(x) and numbers L, c, and ε>0. In each case, find the largest open interval about c on which the inequality |f(x)−L| <ε holds. Then give a value for δ>0 such that for all x satisfying 0 < |x−c| < δ, the inequality |f(x)−L| < ε holds.


f(x) = 1/x, L = 1/4, c = 4, ε = 0.05

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