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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3:00 minutes

Problem 58

Textbook Question

Evaluate each limit.

lim x→0+ 1−cos^2x / sin x

Verified step by step guidance

Step 1: Recognize that the expression $1 - \cos^2 x$ can be rewritten using the Pythagorean identity $\sin^2 x + \cos^2 x = 1$. Therefore, $1 - \cos^2 x = \sin^2 x$.

Step 2: Substitute $\sin^2 x$ for $1 - \cos^2 x$ in the limit expression. The limit now becomes $\lim_{{x \to 0^+}} \frac{\sin^2 x}{\sin x}$.

Step 3: Simplify the expression $\frac{\sin^2 x}{\sin x}$ by canceling one $\sin x$ from the numerator and the denominator. This simplifies to $\sin x$.

Step 4: Evaluate the limit $\lim_{{x \to 0^+}} \sin x$. Since $\sin x$ is continuous at $x = 0$, you can directly substitute $x = 0$ into $\sin x$.

Step 5: Conclude the evaluation by noting that the limit of $\sin x$ as $x$ approaches 0 from the positive side is simply $\sin(0)$.

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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. In this case, we are interested in the limit of the expression as x approaches 0 from the positive side (0+). Understanding limits is crucial for evaluating functions that may not be directly computable at specific points.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are periodic functions that relate angles to ratios of sides in right triangles. In the given limit, we encounter cos^2(x) and sin(x), which require knowledge of their properties and behaviors, especially near critical points like x = 0, where they can exhibit specific limits and values.

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 in simplifying complex limit problems involving trigonometric functions.

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Textbook Question

The hyperbolic cosine function, denoted $\cosh (x)$ , is used to model the shape of a hanging cable (a telephone wire, for example). It is defined as $\cosh (x) = \frac{e^{x} + e^{- x}}{2}$ .