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 50

Textbook Question

Find the limits in Exercises 49–52. Write ∞ or −∞ where appropriate.

lim x→(−π/2)⁺ sec x

Verified step by step guidance

Understand the function: The secant function, sec(x), is defined as 1/cos(x). As x approaches a certain value, we need to consider the behavior of the cosine function.

Identify the point of interest: We are looking at the limit as x approaches -π/2 from the right (denoted as x→(-π/2)⁺). This means we are considering values of x that are slightly greater than -π/2.

Analyze the cosine function: At x = -π/2, cos(x) is 0. As x approaches -π/2 from the right, cos(x) approaches 0 from the positive side, meaning cos(x) is positive but very small.

Consider the behavior of sec(x): Since sec(x) = 1/cos(x), as cos(x) approaches 0 from the positive side, sec(x) will become very large. This indicates that sec(x) approaches positive infinity.

Conclude the limit: Based on the analysis, the limit of sec(x) as x approaches -π/2 from the right is ∞.

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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 how functions behave near points of interest, including points of discontinuity or infinity. Evaluating limits is essential for defining derivatives and integrals, which are core components of calculus.

Secant Function

The secant function, denoted as sec(x), is defined as the reciprocal of the cosine function: sec(x) = 1/cos(x). It is important to understand the behavior of sec(x) as x approaches certain values, particularly where cos(x) equals zero, since this leads to undefined values and vertical asymptotes in the secant function.

One-Sided Limits

One-sided limits refer to the value that a function approaches as the input approaches a specific point from one side, either the left (−) or the right (+). In the context of the given limit, evaluating the right-hand limit as x approaches −π/2 helps determine the behavior of sec(x) near this critical point, which is essential for understanding discontinuities in the function.