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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 2.2.64
Chapter 2, Problem 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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Key Concepts

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

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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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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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

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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) .

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Textbook Question

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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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)

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Textbook Question

Limits with trigonometric functions


Find the limits in Exercises 43–50.


lim x→0 tan x

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Textbook Question

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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