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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2:23 minutes

Problem 2.12b

Textbook Question

Graph the function f(x)=e^−x / x(x+2)^2 using a graphing utility. (Experiment with your choice of a graphing window.) Use your graph to determine the following limits.

b. lim x→−2 f(x)

Verified step by step guidance

Identify the function: \( f(x) = \frac{e^{-x}}{x(x+2)^2} \).

Recognize that the limit \( \lim_{x \to -2} f(x) \) involves a point where the denominator becomes zero, indicating a potential vertical asymptote or removable discontinuity.

Analyze the behavior of the function as \( x \) approaches \(-2\) from both the left and the right to determine if the limit exists.

Consider the sign and magnitude of the numerator \( e^{-x} \) and the denominator \( x(x+2)^2 \) as \( x \to -2^- \) and \( x \to -2^+ \).

Use a graphing utility to visualize the function near \( x = -2 \) to confirm the behavior and determine if the limit exists or if there is a vertical asymptote.

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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 of discontinuity or where the function may not be explicitly defined. Evaluating limits is crucial for determining the continuity and differentiability of functions.

Graphing Functions

Graphing functions involves plotting the values of a function on a coordinate system to visualize its behavior. This process can reveal important features such as intercepts, asymptotes, and intervals of increase or decrease. Using a graphing utility allows for experimentation with different viewing windows, which can help in identifying the limits and overall shape of the function.

Asymptotic Behavior

Asymptotic behavior refers to how a function behaves as it approaches a certain point, particularly at infinity or near points of discontinuity. In the context of limits, understanding asymptotic behavior is essential for determining the value of a limit as the input approaches a specific value, such as -2 in this case. It often involves analyzing the function's growth rates and identifying any vertical or horizontal asymptotes.

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Evaluate each limit and justify your answer.

lim x→∞(2x+1x / x)^3

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Evaluate each limit and justify your answer.

lim x→5 ln 6(√x^2−16−3) / 5x−25

Textbook Question

Evaluate each limit and justify your answer.

lim x→0 (x / √16x+1-1)^1/3

Textbook Question

Graph the function f(x)=e^−x / x(x+2)^2 using a graphing utility. (Experiment with your choice of a graphing window.) Use your graph to determine the following limits.

a. lim x→−2^+ f(x)

Textbook Question

Graph the function f(x)=e^−x / x(x+2)^2 using a graphing utility. (Experiment with your choice of a graphing window.) Use your graph to determine the following limits.

d. lim x→0^+ f(x)

Textbook Question

Graph the function f(x)=e^−x / x(x+2)^2 using a graphing utility. (Experiment with your choice of a graphing window.) Use your graph to determine the following limits.

c. lim x→0^− f(x)

Textbook Question

Suppose f(x)→100 and g(x)→0, with g(x)<0 as x→2. Determine lim x→2 f(x) / g(x).

Textbook Question

Which of the following statements are correct? Choose all that apply.

a. lim x→1 1/ (x−1)^2 does not exist

b. lim x→1 1/ (x−1)^2=∞

c. lim x→1 1/(x−1)^2=−∞

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