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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1:06 minutes

Problem 2.3.59

Textbook Question

Find the following limits or state that they do not exist. Assume a, b, c, and k are fixed real numbers.

$\lim_{x \to \infty} x \cos (x)$

Verified step by step guidance

Recognize that the limit involves an oscillating function, $ \cos(x) $, which oscillates between -1 and 1.

Consider the behavior of $ x \cos(x) $ as $ x \to \infty $. The term $ x $ grows without bound, while $ \cos(x) $ continues to oscillate.

Use the Squeeze Theorem to analyze the limit. Since $ -1 \leq \cos(x) \leq 1 $, it follows that $ -x \leq x \cos(x) \leq x $.

Examine the limits of the bounding functions: $ \lim_{x \to \infty} -x = -\infty $ and $ \lim_{x \to \infty} x = \infty $.

Conclude that since the bounding functions tend to $ -\infty $ and $ \infty $, the limit $ \lim_{x \to \infty} x \cos(x) $ does not exist.

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Key Concepts

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

Limits at Infinity

Limits at infinity involve evaluating the behavior of a function as the input approaches infinity. This concept is crucial for understanding how functions behave in extreme cases, particularly for determining whether they approach a finite value, diverge, or oscillate. In this context, we analyze the limit of the function as x approaches infinity to ascertain its long-term behavior.

Oscillatory Behavior

Oscillatory behavior refers to functions that do not settle at a single value as their input increases but instead fluctuate between values. The cosine function, for example, oscillates between -1 and 1. When combined with a term that grows without bound, such as x in this limit, it can lead to indeterminate forms, necessitating careful analysis to determine the limit's existence.

Squeeze Theorem

The Squeeze Theorem is a method used to find limits of functions that are difficult to evaluate directly. It states that if a function is 'squeezed' between two other functions that converge to the same limit, then the squeezed function must also converge to that limit. This theorem is particularly useful in cases where oscillatory functions are involved, as it can help establish the limit's behavior by bounding it.