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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1:40 minutes

Problem 22

Textbook Question

Finding Limits
In Exercises 9–24, find the limit or explain why it does not exist.

lim x →π cos² (x― tan x)

Verified step by step guidance

First, understand the expression for which you need to find the limit: \( \lim_{x \to \pi} \cos^2(x - \tan x) \). This involves the cosine function squared, evaluated at \( x - \tan x \).

Next, evaluate the behavior of \( x - \tan x \) as \( x \) approaches \( \pi \). Since \( \tan x \) is undefined at \( x = \pi \), consider the behavior of \( \tan x \) near \( \pi \).

Recognize that \( \tan x \) approaches 0 as \( x \) approaches \( \pi \) from either side, because \( \tan(\pi) = 0 \). Therefore, \( x - \tan x \) approaches \( \pi \) as \( x \to \pi \).

Substitute the limit of \( x - \tan x \) into the cosine function: \( \cos^2(x - \tan x) \to \cos^2(\pi) \). Recall that \( \cos(\pi) = -1 \), so \( \cos^2(\pi) = (-1)^2 = 1 \).

Conclude that the limit of the original expression is \( 1 \), as the squared cosine of \( \pi \) is \( 1 \). Thus, \( \lim_{x \to \pi} \cos^2(x - \tan x) = 1 \).

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Key Concepts

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

Limits

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 the function's value at points where it may not be explicitly defined. For example, limits are essential for evaluating functions at points of discontinuity or for determining the behavior of functions at infinity.

Trigonometric Functions

Trigonometric functions, such as cosine and tangent, are periodic functions that relate angles to ratios of sides in right triangles. Understanding these functions is crucial for evaluating limits involving angles, especially when they approach specific values like π. The behavior of these functions near their critical points can significantly affect the limit's outcome.

Continuity

Continuity refers to a property of functions where small changes in the input result in small changes in the output. A function is continuous at a point if the limit as the input approaches that point equals the function's value at that point. This concept is vital when finding limits, as it helps determine whether a limit exists and if it can be evaluated directly.