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

Finding Limits Algebraically

Calculus

1. Limits and Continuity

Finding Limits Algebraically

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2:05 minutes

Problem 2.43

Textbook Question

Limits and Infinity

Find the limits in Exercises 37–46.

sin x
lim ------------- ( If you have a grapher, try graphing
x→∞ |x| the function for ―5 ≤ x ≤ 5 ) .

Verified step by step guidance

Identify the function for which you need to find the limit: \( \frac{\sin x}{|x|} \).

Understand that as \( x \to \infty \), the absolute value \( |x| \) behaves like \( x \) because \( x \) is positive.

Recognize that the sine function, \( \sin x \), oscillates between -1 and 1 for all real numbers \( x \).

Consider the behavior of the fraction \( \frac{\sin x}{x} \) as \( x \to \infty \). Since \( \sin x \) is bounded and \( x \) grows without bound, the fraction approaches zero.

Conclude that the limit of \( \frac{\sin x}{|x|} \) as \( x \to \infty \) is 0, because the numerator is bounded while the denominator increases indefinitely.

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Key Concepts

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

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. They help in understanding the behavior of functions at specific points, including points of discontinuity or infinity. In this context, evaluating the limit as x approaches infinity allows us to analyze the long-term behavior of the function sin(x)/|x|.

Behavior of Functions at Infinity

When analyzing limits as x approaches infinity, we assess how a function behaves as its input grows without bound. This often involves determining whether the function approaches a finite value, diverges to infinity, or oscillates. For the function sin(x)/|x|, understanding its behavior as x becomes very large is crucial for finding the limit.

Trigonometric Functions

Trigonometric functions, such as sine, exhibit periodic behavior, oscillating between fixed values. The function sin(x) oscillates between -1 and 1, which is important when considering its limit in conjunction with |x|. This periodic nature influences the overall limit of the function sin(x)/|x| as x approaches infinity, as the oscillation will be divided by an increasingly large denominator.