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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1:30 minutes

Problem 2.2.50

Textbook Question

Limits with trigonometric functions

Find the limits in Exercises 43–50.

limx→0 √(7 + sec²x)

Verified step by step guidance

Step 1: Understand the problem. We need to find the limit of the function \( \sqrt{7 + \sec^2 x} \) as \( x \) approaches 0.

Step 2: Recall the definition of the secant function. The secant function is defined as \( \sec x = \frac{1}{\cos x} \). Therefore, \( \sec^2 x = \frac{1}{\cos^2 x} \).

Step 3: Substitute \( \sec^2 x \) in the original function. The expression becomes \( \sqrt{7 + \frac{1}{\cos^2 x}} \).

Step 4: Evaluate the limit as \( x \to 0 \). As \( x \to 0 \), \( \cos x \to 1 \), so \( \cos^2 x \to 1 \). Therefore, \( \frac{1}{\cos^2 x} \to 1 \).

Step 5: Simplify the expression inside the square root. As \( x \to 0 \), the expression inside the square root becomes \( 7 + 1 = 8 \). Thus, the limit of the function is \( \sqrt{8} \).

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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 in calculus, representing the value that a function approaches as the input approaches a certain point. They are essential for understanding continuity, derivatives, and integrals. In this context, evaluating the limit as x approaches 0 helps determine the behavior of the function near that point.

Trigonometric Functions

Trigonometric functions, such as secant (sec), are periodic functions that relate angles to ratios of sides in right triangles. The secant function is defined as the reciprocal of the cosine function. Understanding how these functions behave, especially near specific angles like 0, is crucial for evaluating limits involving them.

Secant Function Behavior

The secant function, sec(x), approaches infinity as x approaches odd multiples of π/2, where the cosine function is zero. However, as x approaches 0, sec(x) approaches 1 since cos(0) = 1. This behavior is important for evaluating the limit of the expression √(7 + sec²x) as x approaches 0.