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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1:54 minutes

Problem 33a

Textbook Question

Determine whether the following statements are true and give an explanation or counterexample.

a. The value of $\lim_{x \to 3} \frac{x^{2} - 9}{x - 3}$ does not exist.

Verified step by step guidance

Identify the limit expression: $ \lim_{x \to 3} \frac{x^2 - 9}{x - 3} $.

Notice that direct substitution of $ x = 3 $ results in a $ \frac{0}{0} $ indeterminate form.

Factor the numerator: $ x^2 - 9 = (x - 3)(x + 3) $.

Rewrite the limit expression: $ \lim_{x \to 3} \frac{(x - 3)(x + 3)}{x - 3} $.

Cancel the common factor $ x - 3 $ in the numerator and denominator, then evaluate the limit of the simplified expression.

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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 how functions behave near points of interest, including points where they may not be defined. For example, the limit of a function can exist even if the function itself does not take a value at that point.

Indeterminate Forms

Indeterminate forms arise in calculus when evaluating limits that do not lead to a clear value, such as 0/0 or ∞/∞. These forms require further analysis, often using algebraic manipulation or L'Hôpital's Rule, to resolve the limit. Recognizing these forms is crucial for determining the existence of limits in complex expressions.

Factoring and Simplifying Expressions

Factoring and simplifying expressions is a technique used to rewrite complex algebraic expressions in a more manageable form. In the context of limits, this often involves canceling common factors to eliminate indeterminate forms. For instance, the expression (x^2 - 9)/(x - 3) can be factored to (x - 3)(x + 3)/(x - 3), allowing for simplification and easier limit evaluation.

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Related Practice

Textbook Question

Let $g\left(x\right)=\frac{x^3-4x}{8\left|x-2\right|}$ . <IMAGE>

Calculate $g\left(x\right)$ for each value of $x$ in the following table.

Textbook Question

Let $g\left(x\right)=\frac{x^3-4x}{8\left|x-2\right|}$ . <IMAGE>

Make a conjecture about the values of ${\displaystyle\lim_{x\to2^{-}}g\left(x\right)}$ , ${\displaystyle\lim_{x\to2^{+}}g\left(x\right)}$ , and ${\displaystyle\lim_{x\to2}g\left(x\right)}$ or state that they do not exist.

Textbook Question

Use a graph of f to estimate $\lim_{x \to a} f (x)$ or to show that the limit does not exist. Evaluate f(x) near $x = a$ to support your conjecture.

$f (x) = \frac{x - 2}{\ln ∣ x - 2 ∣}$ ; $a equals 2$

Textbook Question

Use a graph of f to estimate $\lim_{x \to a} f (x)$ or to show that the limit does not exist. Evaluate f(x) near $x = a$ to support your conjecture.

$f (x) = \frac{1 - \cos (2 x - 2)}{{(x - 1)}^{2}}; a = 1$

Textbook Question

Determine whether the following statements are true and give an explanation or counterexample.

d. $\lim_{x \to 0} \sqrt{x}$ . (Hint: Graph y=√x)

Textbook Question

Determine whether the following statements are true and give an explanation or counterexample.

e. $\lim_{x \to \frac{π}{2}} \cot x = 0$ . (Hint: Graph y=cot x)

Textbook Question

Sketch the graph of a function with the given properties. You do not need to find a formula for the function.

f(2) = 1,lim x→2 f(x) = 3

Textbook Question

Sketch the graph of a function with the given properties. You do not need to find a formula for the function.

p(0) = 2,lim x→0 p(x) = 0,lim x→2 p(x) does not exist, p(2)=lim x→2^+ p(x)=1

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Determine whether the following statements are true and give an explanation or counterexample.a. The value of limx→3x2−9x−3{\displaystyle\lim_{x\to3}}\frac{x^2-9}{x-3} does not exist.

Key Concepts

Limits

Indeterminate Forms

Factoring and Simplifying Expressions

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Determine whether the following statements are true and give an explanation or counterexample.

a. The value of $\lim_{x \to 3} \frac{x^{2} - 9}{x - 3}$ does not exist.