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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2:30 minutes

Problem 36

Textbook Question

Determine the following limits.

lim x→−∞ (e^x cos x +3)

Verified step by step guidance

Identify the behavior of each component of the function as \( x \to -\infty \).

Recognize that \( e^x \to 0 \) as \( x \to -\infty \) because the exponential function decays to zero.

Note that \( \cos x \) oscillates between -1 and 1 for all real numbers, including as \( x \to -\infty \).

Consider the product \( e^x \cos x \). Since \( e^x \to 0 \) and \( \cos x \) is bounded, \( e^x \cos x \to 0 \) as \( x \to -\infty \).

Combine the results: \( \lim_{x \to -\infty} (e^x \cos x + 3) = 0 + 3 = 3 \).

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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 positive or negative infinity. Understanding how functions behave in these scenarios is crucial for determining their end behavior, which can often simplify complex expressions.

Exponential Functions

Exponential functions, such as e^x, grow rapidly as x increases and approach zero as x decreases towards negative infinity. This characteristic is essential for analyzing limits involving exponential terms, particularly when combined with oscillatory functions like cosine.

Trigonometric Functions

Trigonometric functions, like cos x, oscillate between -1 and 1 regardless of the value of x. When evaluating limits that include trigonometric functions, it's important to recognize their bounded nature, which can influence the overall limit when combined with other terms.