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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2:53 minutes

Problem 2.57

Textbook Question

Evaluate each limit.

lim x→0 cos x−1 / sin^2x

Verified step by step guidance

Step 1: Recognize that the limit involves an indeterminate form 0/0 as x approaches 0, which suggests the use of L'Hôpital's Rule. L'Hôpital's Rule states that if the limit of f(x)/g(x) as x approaches a value results in an indeterminate form, the limit can be evaluated as the limit of f'(x)/g'(x).

Step 2: Differentiate the numerator and the denominator separately. The derivative of the numerator, cos(x) - 1, is -sin(x). The derivative of the denominator, sin^2(x), is 2sin(x)cos(x) using the chain rule.

Step 3: Apply L'Hôpital's Rule by taking the limit of the new fraction formed by the derivatives: lim x→0 [-sin(x)] / [2sin(x)cos(x)].

Step 4: Simplify the expression. Notice that the sin(x) terms in the numerator and denominator can be canceled out, resulting in the limit of -1 / [2cos(x)] as x approaches 0.

Step 5: Evaluate the simplified limit by substituting x = 0 into the expression -1 / [2cos(x)]. Since cos(0) = 1, the expression simplifies further.

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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 behavior near points of interest, including points where the function may not be explicitly defined. Evaluating limits is essential for determining continuity, derivatives, and integrals.

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) as x approaches a point yields an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator separately. This rule simplifies the process of finding limits in complex expressions.

Trigonometric Limits

Trigonometric limits involve the evaluation of limits that include trigonometric functions, such as sine and cosine. These limits often require special techniques or known limit values, such as lim x→0 (sin x)/x = 1. Understanding the behavior of trigonometric functions near specific points is crucial for solving problems involving limits in calculus.

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Related Practice

Textbook Question

Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any horizontal asymptotes.

f(x)=cos x+2√x / √x.

Textbook Question

Find the vertical asymptotes. For each vertical asymptote x=a, analyze lim x→a^− f(x) and lim x→a^+f(x).

f(x)=cos x+2√x / √x.

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 $\cosh (x) = \frac{e^{x} + e^{- x}}{2}$ .