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

Previous problemNext problem

2:15 minutes

Problem 2.2.47

Textbook Question

Limits with trigonometric functions

Find the limits in Exercises 43–50.

lim x→0 tan x

Verified step by step guidance

Recognize that the limit involves the trigonometric function tan(x) as x approaches 0.

Recall the identity for tangent: tan(x) = sin(x) / cos(x). This will help in simplifying the expression.

Consider the limit: lim x→0 tan(x) = lim x→0 (sin(x) / cos(x)).

Use the fact that as x approaches 0, sin(x) approaches 0 and cos(x) approaches 1. This simplifies the expression.

Apply the limit: lim x→0 (sin(x) / cos(x)) = (lim x→0 sin(x)) / (lim x→0 cos(x)) = 0 / 1.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

Play a video:

Was this helpful?

Key Concepts

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

Limits

Limits are fundamental concepts in calculus that describe the behavior of a function as its input approaches a certain value. They help in understanding the function's behavior near points of interest, including points of discontinuity or where the function is not explicitly defined. Evaluating limits is essential for defining derivatives and integrals.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, relate angles to ratios of sides in right triangles. They are periodic functions that play a crucial role in calculus, especially when dealing with limits, derivatives, and integrals involving angles. Understanding their properties and behaviors is key to solving limit problems involving these functions.

L'Hôpital's Rule

L'Hôpital's Rule is a method used to evaluate limits that result in indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of f(x)/g(x) leads to 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.