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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2:46 minutes

Problem 2.4.2

Textbook Question

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

Verified step by step guidance

Step 1: Understand the notation: The expression $ \lim_{x \to a} f(x) = \infty $ is a way to describe the behavior of a function $ f(x) $ as the variable $ x $ approaches a specific value $ a $.

Step 2: Interpret the meaning: This notation means that as $ x $ gets closer and closer to $ a $, the values of $ f(x) $ increase without bound. In other words, $ f(x) $ becomes arbitrarily large.

Step 3: Consider the context: This type of limit is often used to describe vertical asymptotes in graphs, where the function grows indefinitely as it approaches a certain point.

Step 4: Visualize the concept: Imagine a graph where as you move along the x-axis towards $ a $, the y-values (outputs of $ f(x) $) shoot upwards towards infinity, never settling at a finite number.

Step 5: Relate to real-world examples: Think of scenarios like the speed of an object approaching the speed of light, where the energy required becomes infinitely large as the speed approaches a certain limit.

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Key Concepts

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

Limit of a Function

The limit of a function describes the behavior of that function as the input approaches a certain value. Specifically, lim x→a f(x) indicates the value that f(x) approaches as x gets infinitely close to a. Understanding limits is fundamental in calculus, as they form the basis for defining continuity, derivatives, and integrals.

Infinite Limits

An infinite limit occurs when the value of a function increases without bound as the input approaches a specific point. In the expression lim x→a f(x) = ∞, it signifies that as x approaches 'a', the function f(x) grows larger and larger, indicating that it does not settle at a finite value. This concept is crucial for analyzing vertical asymptotes in graphs.

Vertical Asymptotes

Vertical asymptotes are lines that a graph approaches but never touches or crosses, typically occurring at points where a function tends toward infinity. When lim x→a f(x) = ∞, it suggests that there is a vertical asymptote at x = a, indicating a significant change in the function's behavior near that point. Recognizing vertical asymptotes helps in understanding the overall shape and limits of functions.

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Related Practice

Textbook Question

Determine the following limits.

a. lim x→2^+ x^2 − 4x + 3 / (x − 2)^2

Textbook Question

Determine the following limits.

b. lim t→−2^− t^3 − 5t^2 + 6t / t^4 − 4t^2

Textbook Question

Find all vertical asymptotes $x = a$ of the following functions. For each value of $a$ , determine $\lim_{x \to a^{+}} f (x)$ , $\lim_{x \to a^{-}} f (x)$ , and $\lim_{x \to a} f (x)$ .

$f (x) = \frac{\cos (x)}{x^{2} + 2 x}$

Textbook Question

Find all vertical asymptotes $x=a$ of the following functions. For each value of $a$ , determine ${\displaystyle\lim_{x\to a^{+}}}f\left(x\right)$ , ${\displaystyle\lim_{x\to a^{-}}}f\left(x\right)$ , and ${\displaystyle\lim_{x\to a}}f\left(x\right)$ .

$f (x) = \frac{x + 1}{x^{3} - 4 x^{2} + 4 x}$

Textbook Question

Use the graph of f(x) = x / (x² − 2x − 3)² to determine lim x→−1 f(x) and lim x→3 f(x).

Textbook Question

Analyze the following limits and find the vertical asymptotes of f(x) = (x − 5) / (x² − 25).

lim x → -5^- f(x)

Textbook Question

Analyze the following limits and find the vertical asymptotes of f(x) =(x − 5) / (x² − 25).

lim x→−5⁺ f(x)

Textbook Question

Analyze the following limits and find the vertical asymptotes of f(x) = (x + 7) / (x⁴ − 49x²).

lim x → -7 f(x)

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