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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2:24 minutes

Problem 3.5.2

Textbook Question

How is lim x🠂0 sin x/x used in this section?

Verified step by step guidance

Step 1: Recognize that the limit \( \lim_{x \to 0} \frac{\sin x}{x} \) is a fundamental limit in calculus, often used in the study of derivatives and integrals involving trigonometric functions.

Step 2: Understand that this limit evaluates to 1, which is crucial for solving problems involving small angle approximations and for finding derivatives of trigonometric functions.

Step 3: Apply this limit when dealing with expressions that can be rewritten in the form \( \frac{\sin x}{x} \) as \( x \to 0 \). This often involves algebraic manipulation to match the form.

Step 4: Use this limit to derive the derivative of \( \sin x \), which is \( \cos x \), by considering the definition of the derivative and applying the limit.

Step 5: Recognize that this limit is also used in L'Hôpital's Rule, which helps evaluate indeterminate forms like \( \frac{0}{0} \) by differentiating the numerator and the denominator.

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Key Concepts

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

Limit of a Function

The limit of a function describes the value that a function approaches as the input approaches a certain point. In calculus, limits are fundamental for defining continuity, derivatives, and integrals. The expression lim x→0 sin x/x is a classic limit that evaluates the behavior of the sine function as x approaches zero, which is crucial for understanding the function's properties near that point.

Sine Function Behavior

The sine function, denoted as sin(x), is a periodic function that oscillates between -1 and 1. Its behavior near zero is particularly important in calculus, as it helps in approximating values and understanding the function's growth. The limit lim x→0 sin x/x reveals that as x approaches zero, the ratio approaches 1, illustrating the relationship between the sine function and linear approximations.

L'Hôpital's Rule

L'Hôpital's Rule is a method for evaluating limits of indeterminate forms, such as 0/0 or ∞/∞. When faced with such forms, the rule states that the limit of the ratio of two functions can be found by taking the derivative of the numerator and the derivative of the denominator. This rule can be applied to the limit lim x→0 sin x/x, confirming that the limit equals 1 by differentiating both the numerator and denominator.

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Textbook Question

Use Theorem 3.10 to evaluate the following limits.

lim x🠂0 (sin 7x) / 3x

Textbook Question

Use Theorem 3.10 to evaluate the following limits.

lim x🠂0 (tan 5x) / x

Textbook Question

Use Theorem 3.10 to evaluate the following limits.

lim x🠂0 (tan 7x) / (sin x)

Textbook Question

Use Theorem 3.10 to evaluate the following limits.

lim x🠂2 (sin (x-2)) / (x² - 4)

Textbook Question

Another method for proving lim x→0 cos x−1/x = 0 Use the half-angle formula sin²x = 1− cos 2x/2 to prove that lim x→0 cos x−1/x=0.

Textbook Question

{Use of Tech} Computing limits with angles in degrees Suppose your graphing calculator has two functions, one called sin x, which calculates the sine of x when x is in radians, and the other called s(x), which calculates the sine of x when x is in degrees.

b. Evaluate lim x→0 s(x) / x. Verify your answer by estimating the limit on your calculator.

Textbook Question

Finding Limits

In Exercises 25–28, find the limit of g(x) as x approaches the indicated value.

lim (4g(x))¹/³ = 2

x →0

Textbook Question

Finding Limits

In Exercises 25–28, find the limit of g(x) as x approaches the indicated value.

5 ―x²

lim ------------- = 0

x → ―2 (√g(x))