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

Introduction to Limits

Calculus

1. Limits and Continuity

Introduction to Limits

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2:16 minutes

Problem 1

Textbook Question

Explain the meaning of lim x→−∞ f(x)=10.

Verified step by step guidance

Step 1: Understand the notation: The expression $ \lim_{x \to -\infty} f(x) = 10 $ is read as 'the limit of $ f(x) $ as $ x $ approaches negative infinity is 10.'

Step 2: Conceptualize the behavior: This means that as $ x $ becomes very large in the negative direction (i.e., $ x $ goes to negative infinity), the values of the function $ f(x) $ get closer and closer to 10.

Step 3: Visualize the graph: Imagine the graph of $ f(x) $. As you move left along the x-axis towards negative infinity, the y-values (outputs of $ f(x) $) approach the horizontal line $ y = 10 $.

Step 4: Consider the horizontal asymptote: The line $ y = 10 $ can be considered a horizontal asymptote of the function $ f(x) $ as $ x \to -\infty $. This means the graph of $ f(x) $ gets closer to this line but may not necessarily touch or cross it.

Step 5: Relate to real-world scenarios: In practical terms, this limit could represent a situation where a quantity stabilizes at a certain value (10 in this case) as time or another variable decreases without bound.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limit at Infinity

The limit at infinity describes the behavior of a function as the input approaches infinity or negative infinity. Specifically, lim x→−∞ f(x)=10 indicates that as x decreases without bound, the values of f(x) approach 10. This concept is crucial for understanding how functions behave in extreme cases.

Horizontal Asymptote

A horizontal asymptote is a horizontal line that a graph approaches as x approaches infinity or negative infinity. In this case, the line y=10 serves as a horizontal asymptote for the function f(x) as x approaches negative infinity, indicating that f(x) stabilizes around this value rather than diverging.

Function Behavior

Understanding function behavior involves analyzing how a function changes as its input varies. The statement lim x→−∞ f(x)=10 suggests that for very large negative values of x, the function f(x) behaves consistently, converging to the value 10, which is essential for predicting the function's long-term trends.

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Textbook Question

Determine the following limits at infinity.

lim x→∞ (3+10/x^2)

Textbook Question

Determine the following limits at infinity.

lim x→∞ (5 + 1/x +10/x^2)

Textbook Question

Determine the following limits at infinity.

lim t→∞ et,lim t→−∞ e^t,and lim t→∞ e^−t

Textbook Question

Sketch a possible graph of a function f that satisfies all of the given conditions. Be sure to identify all vertical and horizontal asymptotes.

$f (- 1) = - 2$ , $f of 1 equals 2$ , $f of 0 equals 0$ , $\lim_{x \to \infty} f (x) = 1$ , $\lim_{x \to - \infty} f (x) = - 1$

Textbook Question

Estimate lim θ→0 sin 2θ / sin θ using the graph in part (a).

Textbook Question

a. Estimate lim x→π/4 cos 2x / cos x − sin x by making a table of values of cos 2x / cos x − sin x for values of x approaching π/4. Round your estimate to four digits.

Textbook Question

Evaluate lim x→∞ f(x) and lim x→−∞ f(x) sing the figure. <IMAGE>

Textbook Question

Sketch a graph of f and use it to make a conjecture about the values of f(a), lim x→a^−f(x),lim x→a^+f(x), and lim x→a f(x) or state that they do not exist.

f(x) = {x^2+1 if x≤−1

3 if x>−1; a=−1

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