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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 2.2.64
Capitolo 2, Problema 2.2.64

Using the Sandwich Theorem


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

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Identify the functions involved in the inequality: f(x) = 2 - x² and h(x) = 2cos(x). We know that f(x) ≤ g(x) ≤ h(x) for all x.
Evaluate the limit of f(x) as x approaches 0: lim(x→0) (2 - x²). Since x² approaches 0 as x approaches 0, the limit of f(x) is 2.
Evaluate the limit of h(x) as x approaches 0: lim(x→0) 2cos(x). Since cos(0) = 1, the limit of h(x) is 2.
Apply the Sandwich Theorem (also known as the Squeeze Theorem), which states that if f(x) ≤ g(x) ≤ h(x) and lim(x→a) f(x) = lim(x→a) h(x) = L, then lim(x→a) g(x) = L.
Conclude that since both limits of f(x) and h(x) as x approaches 0 are equal to 2, by the Sandwich Theorem, lim(x→0) g(x) = 2.

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Sandwich Theorem

The Sandwich Theorem, also known as the Squeeze Theorem, states that if a function g(x) is bounded by two other functions f(x) and h(x) such that f(x) ≤ g(x) ≤ h(x) for all x in an interval, and if the limits of f(x) and h(x) as x approaches a point are equal, then the limit of g(x) as x approaches that point is also equal to that limit.
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Fundamental Theorem of Calculus Part 1

Limit of a Function

The limit of a function describes the value that the function approaches as the input approaches a certain point. In this context, we are interested in finding the limit of g(x) as x approaches 0, which requires evaluating the behavior of g(x) near that point based on the bounding functions.
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Limits of Rational Functions: Denominator = 0

Bounding Functions

Bounding functions are the functions that enclose another function within a specific interval. In this case, 2−x² and 2cosx serve as the bounding functions for g(x). Understanding their limits as x approaches 0 is crucial for applying the Sandwich Theorem to find the limit of g(x).
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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

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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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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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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.


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Limits with trigonometric functions


Find the limits in Exercises 43–50.


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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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