Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

1. Limits and Continuity

Introduction to Limits

Calculus

1. Limits and Continuity

Introduction to Limits

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2:33 minutes

Problem 2.2.64

Textbook Question

Using the Sandwich Theorem

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

Verified step by step guidance

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.

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.

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

Limits of Average Rates of Change

Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form limh→0 (f(x+h) − f(x)) / h occur frequently in calculus. In Exercises 57–62, evaluate this limit for the given value of x and function f.

f(x) = x², x = -2