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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2:19 minutes

Problem 2.35

Textbook Question

Determine the following limits.

lim x→∞ sin x / e^x

Verified step by step guidance

Step 1: Understand the problem. We need to find the limit of the function \( \frac{\sin x}{e^x} \) as \( x \) approaches infinity.

Step 2: Analyze the behavior of the numerator and the denominator separately. The function \( \sin x \) oscillates between -1 and 1 for all \( x \).

Step 3: Consider the behavior of the denominator \( e^x \). As \( x \) approaches infinity, \( e^x \) grows exponentially and becomes very large.

Step 4: Apply the Squeeze Theorem. Since \( -1 \leq \sin x \leq 1 \), we have \( -\frac{1}{e^x} \leq \frac{\sin x}{e^x} \leq \frac{1}{e^x} \).

Step 5: Evaluate the limits of the bounding functions. As \( x \to \infty \), both \( \frac{1}{e^x} \to 0 \) and \( -\frac{1}{e^x} \to 0 \). By the Squeeze Theorem, \( \lim_{x \to \infty} \frac{\sin x}{e^x} = 0 \).

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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. In this context, we analyze how the function behaves as x becomes very large, which can reveal whether the function approaches a specific value, diverges, or oscillates.

Behavior of Sinusoidal Functions

The sine function oscillates between -1 and 1 for all real numbers. This bounded behavior is crucial when evaluating limits involving sine, as it indicates that despite the oscillation, the overall contribution of sin(x) becomes negligible compared to other functions that grow without bound, such as exponential functions.

Exponential Growth

Exponential functions, like e^x, grow significantly faster than polynomial or sinusoidal functions as x approaches infinity. This rapid growth is key in limit problems, as it often leads to the conclusion that terms involving e^x will dominate the behavior of the limit, driving the overall limit towards zero when combined with bounded functions.

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