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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3:09 minutes

Problem 2.4.28

Textbook Question

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limt→0 2t / tan t

Verified step by step guidance

Recognize that the limit involves a trigonometric function, specifically tan(t), which can be expressed in terms of sin(t) and cos(t). Recall that tan(t) = sin(t) / cos(t).

Rewrite the expression 2t / tan(t) as 2t / (sin(t) / cos(t)), which simplifies to 2t * (cos(t) / sin(t)).

This can be further simplified to (2t * cos(t)) / sin(t).

To apply the known limit lim(θ→0) sin(θ) / θ = 1, rewrite the expression as (2 * cos(t)) * (t / sin(t)).

Recognize that as t approaches 0, t / sin(t) approaches 1, and cos(t) approaches cos(0) = 1. Therefore, the limit can be evaluated by considering the product of these limits: 2 * 1 * 1.

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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 understanding continuity, derivatives, and integrals. The notation lim(x→a) f(x) indicates the limit of f(x) as x approaches a. Evaluating limits often involves techniques such as substitution, factoring, or using special limit properties.

Trigonometric Limits

Trigonometric limits, particularly those involving sine and tangent functions, are essential in calculus. A key result is lim(θ→0) sin(θ)/θ = 1, which helps evaluate limits involving sine and tangent as they approach zero. This result is crucial for simplifying expressions and solving problems that involve trigonometric functions, especially in the context of derivatives and integrals.

L'Hôpital's Rule

L'Hôpital's Rule is a method for evaluating limits that result in indeterminate forms like 0/0 or ∞/∞. It states that if the limit of f(x)/g(x) as x approaches a yields an indeterminate form, then the limit can be found by taking the derivative of the numerator and the derivative of the denominator. This rule simplifies the process of finding limits, especially when dealing with complex functions.

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Related Practice

Textbook Question

[Technology Exercise] Grinding engine cylinders Before contracting to grind engine cylinders to a cross-sectional area of 9in², you need to know how much deviation from the ideal cylinder diameter of c = 3.385in. you can allow and still have the area come within 0.01in² of the required 9in². To find out, you let A=π(x/2)² and look for the largest interval in which you must hold x to make |A − 9| ≤ 0.01. What interval do you find?

Textbook Question

Finding Limits Graphically

Which of the following statements about the function y = f(x) graphed here are true, and which are false?

a. limx→−1+ f(x) = 1

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.

limx→0 (x² − x + sin x) / 2x

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)

Textbook Question

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.