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

Introduction to Limits

Calculus

1. Limits and Continuity

Introduction to Limits

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2:43 minutes

Problem 2.4.35

Textbook Question

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limt→0 sin(1 − cos t) / (1 − cos t)

Verified step by step guidance

Recognize that the limit involves the expression sin(1 - cos t) / (1 - cos t) as t approaches 0. This is a form that can be related to the standard limit lim(θ→0) sin(θ)/θ = 1.

To use the standard limit, we need to manipulate the expression to match the form sin(θ)/θ. Notice that both the numerator and the denominator have the same expression (1 - cos t).

Apply the standard limit by setting θ = 1 - cos t. As t approaches 0, cos t approaches 1, making θ approach 0.

Rewrite the limit in terms of θ: lim(t→0) sin(1 - cos t) / (1 - cos t) becomes lim(θ→0) sin(θ)/θ.

Since lim(θ→0) sin(θ)/θ = 1, conclude that the original limit also equals 1.

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Key Concepts

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

Limit Definition

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near points of interest, particularly where they may not be explicitly defined. The notation lim x→a f(x) indicates the value that f(x) approaches as x gets closer to a.

Trigonometric Limits

Trigonometric limits, such as lim θ→0 sin θ / θ = 1, are specific cases that often arise in calculus. This particular limit is crucial for evaluating limits involving sine and cosine functions, especially when they appear in indeterminate forms. Understanding this limit allows for simplification of expressions involving trigonometric functions as they approach zero.

L'Hôpital's Rule

L'Hôpital's Rule is a method used to evaluate limits of indeterminate forms like 0/0 or ∞/∞. It states that if the limit of f(x)/g(x) results in an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator. This rule is particularly useful when dealing with limits involving trigonometric functions and can simplify complex limit problems.

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Related Practice

Textbook Question

Slope of a Curve at a Point

In Exercises 7–18, use the method in Example 3 to find (a) the slope of the curve at the given point P, and (b) an equation of the tangent line at P.

y=7−x², P(2,3)

Textbook Question

Limits of Average Rates of Change

Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form limh→0 (f(x+h) − f(x)) / h occur frequently in calculus. In Exercises 57–62, evaluate this limit for the given value of x and function f.

f(x) = x², x = 1

Textbook Question

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limt→0 2t / tan t

Textbook Question

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limx→0 (x² − x + sin x) / 2x

Textbook Question

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.