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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2:21 minutes

Problem 27a

Textbook Question

Determine the following limits.

a. lim x→2^+ x^2 − 4x + 3 / (x − 2)^2

Verified step by step guidance

Step 1: Identify the type of limit problem. This is a one-sided limit as $ x $ approaches 2 from the right ($ x \to 2^+ $).

Step 2: Substitute $ x = 2 $ into the function $ \frac{x^2 - 4x + 3}{(x - 2)^2} $ to check if it results in an indeterminate form. Substituting gives $ \frac{2^2 - 4 \times 2 + 3}{(2 - 2)^2} = \frac{4 - 8 + 3}{0} = \frac{-1}{0} $, indicating a division by zero.

Step 3: Analyze the behavior of the numerator and denominator as $ x \to 2^+ $. The numerator $ x^2 - 4x + 3 $ simplifies to $ (x - 1)(x - 3) $. As $ x \to 2^+ $, $ x - 1 \to 1 $ and $ x - 3 \to -1 $, so the numerator approaches $ 1 \times -1 = -1 $.

Step 4: Consider the denominator $ (x - 2)^2 $. As $ x \to 2^+ $, $ x - 2 \to 0^+ $, so $ (x - 2)^2 \to 0^+ $.

Step 5: Determine the limit by combining the behavior of the numerator and denominator. Since the numerator approaches $-1$ and the denominator approaches $0^+$, the limit approaches $-\infty$.

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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. They help in understanding the behavior of functions near specific points, including points of discontinuity or indeterminate forms. Evaluating limits is crucial for defining derivatives and integrals.

Indeterminate Forms

Indeterminate forms occur when direct substitution in a limit leads to expressions like 0/0 or ∞/∞, which do not provide clear information about the limit's value. In such cases, techniques like factoring, rationalizing, or applying L'Hôpital's Rule are used to resolve these forms and find the limit.

Continuous Functions

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. Understanding continuity is essential when evaluating limits, as it allows for the direct substitution of values in many cases, simplifying the limit evaluation process.

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Textbook Question

The graph of ℎ in the figure has vertical asymptotes at x=−2 and x=3. Analyze the following limits. <IMAGE>

lim x→−2 h(x)

Textbook Question

Determine the following limits.

a. lim x→−2^+ (x − 4) / x(x + 2)

Textbook Question

Determine the following limits.

b. lim x→−2^− (x − 4) / x(x + 2)

Textbook Question

Determine the following limits.

c. lim x→−2 (x − 4) / x(x + 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 ${\displaystyle\lim_{x\to a}}f\left(x\right)$ .