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

4. Applications of Derivatives

Differentials

Calculus

4. Applications of Derivatives

Differentials

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2:58 minutes

Problem 4.7.22

Textbook Question

Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→ 0 (eˣ - 1) / (x² + 3x)

Verified step by step guidance

First, substitute x = 0 into the limit expression to check if it results in an indeterminate form. Substituting gives (e^0 - 1) / (0^2 + 3*0) = 0/0, which is an indeterminate form.

Since the limit is in the indeterminate form 0/0, we can apply l'Hôpital's Rule. This rule states that if the limit of f(x)/g(x) as x approaches a value results in 0/0 or ∞/∞, then the limit is the same as the limit of f'(x)/g'(x) as x approaches that value, provided the derivatives exist.

Differentiate the numerator and the denominator separately. The derivative of the numerator, e^x - 1, is e^x. The derivative of the denominator, x² + 3x, is 2x + 3.

Apply l'Hôpital's Rule by taking the limit of the new expression: lim_x→0 (e^x) / (2x + 3).

Substitute x = 0 into the new expression to evaluate the limit: (e^0) / (2*0 + 3) = 1/3. Thus, the limit is 1/3.

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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. Understanding limits is crucial for evaluating expressions that may be indeterminate, such as 0/0 or ∞/∞. In this question, we need to analyze the behavior of the function as x approaches 0 to determine the limit.

l'Hôpital's Rule

l'Hôpital's Rule is a method used to evaluate limits that result in indeterminate forms like 0/0 or ∞/∞. The rule states that if the limit of f(x)/g(x) yields an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator separately. This technique simplifies the evaluation of complex limits, making it particularly useful in this problem.

Exponential Functions

Exponential functions, such as eˣ, are functions where a constant base is raised to a variable exponent. They exhibit unique properties, including the fact that eˣ approaches 1 as x approaches 0. Understanding the behavior of exponential functions is essential for evaluating limits involving them, as they often lead to indeterminate forms that require careful analysis.

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