A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. Video Tutorials on Chapter 2 Limits and Continuity 2.1a Finite Limits Al gebraic and Graphical 2.1 b Finite Limits Squeeze Theorem (Pinch) 2.1 c Finite Limits Graphical and Numerically one and Double sided limits 2.1 d Limits of Piecewise Defined Functions 2.2a Limits Involving Infinity part I. Math 1300: Calculus 1 Section 1.8: Limits Sep 9-10, 2013 Limits We say the limit of f(x) as xapproaches cis Lor lim x!c f(x) = L if f(x) can be made as close as we like to Lby choosing xsufficiently close to (but not equal to) c. Calculus Derivatives Derivative Applications Limits Integrals Integral Applications Riemann Sum Series ODE Multivariable Calculus Laplace Transform Taylor/Maclaurin Series Fourier Series Functions. Stewart Calculus 7e Solutions Chapter 1 Functions and Limits Exercise 1.8. Stewart Calculus 7th Edition Solutions. Chapter 1 Functions and Limits Exercise 1.8 1E. Chapter 1 Functions and Limits Exercise 1.8 2E.
Version #1
The course below follows CollegeBoard's Course and Exam Description. Lessons will begin to appear starting summer 2020.
BC Topics are listed, but there will be no lessons available for SY 2020-2021Unit 0 - Calc Prerequisites (Summer Work)
0.1 Summer Packet Unit 1 - Limits and Continuity
1.1 Can Change Occur at an Instant?
1.2 Defining Limits and Using Limit Notation
1.3 Estimating Limit Values from Graphs
1.4 Estimating Limit Values from Tables
1.5 Determining Limits Using Algebraic Properties
(1.5 includes piecewise functions involving limits)
1.6 Determining Limits Using Algebraic Manipulation
1.7 Selecting Procedures for Determining Limits
(1.7 includes rationalization, complex fractions, and absolute value)
1.8 Determining Limits Using the Squeeze Theorem
1.9 Connecting Multiple Representations of Limits
Mid-Unit Review - Unit 1
1.10 Exploring Types of Discontinuities
1.11 Defining Continuity at a Point
1.12 Confirming Continuity Over an Interval
1.13 Removing Discontinuities
1.14 Infinite Limits and Vertical Asymptotes
1.15 Limits at Infinity and Horizontal Asymptotes
1.16 Intermediate Value Theorem (IVT)
Review - Unit 1 Unit 2 - Differentiation: Definition and Fundamental Properties
2.1 Defining Average and Instantaneous Rate of
Change at a Point
2.2 Defining the Derivative of a Function and Using
Derivative Notation
(2.2 includes equation of the tangent line)
2.3 Estimating Derivatives of a Function at a Point
2.4 Connecting Differentiability and Continuity
2.5 Applying the Power Rule
2.6 Derivative Rules: Constant, Sum, Difference, and
Constant Multiple
(2.6 includes horizontal tangent lines, equation of the
normal line, and differentiability of piecewise)
2.7 Derivatives of cos(x), sin(x), e^x, and ln(x)
2.8 The Product Rule
2.9 The Quotient Rule
2.10 Derivatives of tan(x), cot(x), sec(x), and csc(x)
Review - Unit 2 Unit 3 - Differentiation: Composite, Implicit, and Inverse Functions
3.1 The Chain Rule
3.2 Implicit Differentiation
3.3 Differentiating Inverse Functions
3.4 Differentiating Inverse Trigonometric Functions
3.5 Selecting Procedures for Calculating Derivatives
3.6 Calculating Higher-Order Derivatives
Review - Unit 3
Unit 4 - Contextual Applications of Differentiation
4.1 Interpreting the Meaning of the Derivative in Context
4.2 Straight-Line Motion: Connecting Position, Velocity,
and Acceleration
4.3 Rates of Change in Applied Contexts Other Than
Motion
4.4 Introduction to Related Rates
4.5 Solving Related Rates Problems
4.6 Approximating Values of a Function Using Local
Linearity and Linearization
4.7 Using L'Hopital's Rule for Determining Limits of
Indeterminate Forms
Review - Unit 4 Unit 5 - Analytical Applications of Differentiation
5.1 Using the Mean Value Theorem
5.2 Extreme Value Theorem, Global Versus Local
Extrema, and Critical Points
5.3 Determining Intervals on Which a Function is
Increasing or Decreasing
5.4 Using the First Derivative Test to Determine Relative
Local Extrema
5.5 Using the Candidates Test to Determine Absolute
(Global) Extrema
5.6 Determining Concavity of Functions over Their
Domains
5.7 Using the Second Derivative Test to Determine
Extrema
Mid-Unit Review - Unit 5
5.8 Sketching Graphs of Functions and Their Derivatives
5.9 Connecting a Function, Its First Derivative, and Its
Second Derivative
(5.9 includes a revisit of particle motion and
determining if a particle is speeding up/down.)
5.10 Introduction to Optimization Problems
5.11 Solving Optimization Problems
5.12 Exploring Behaviors of Implicit Relations
Review - Unit 5 Unit 6 - Integration and Accumulation of Change
6.1 Exploring Accumulation of Change
6.2 Approximating Areas with Riemann Sums
6.3 Riemann Sums, Summation Notation, and Definite
Integral Notation
6.4 The Fundamental Theorem of Calculus and
Accumulation Functions
6.5 Interpreting the Behavior of Accumulation Functions
Involving Area
Mid-Unit Review - Unit 6
6.6 Applying Properties of Definite Integrals
6.7 The Fundamental Theorem of Calculus and Definite
Integrals
6.8 Finding Antiderivatives and Indefinite Integrals:
Basic Rules and Notation
6.9 Integrating Using Substitution
6.10 Integrating Functions Using Long Division
and Completing the Square
6.11 Integrating Using Integration by Parts (BC topic)
6.12 Integrating Using Linear Partial Fractions (BC topic)
6.13 Evaluating Improper Integrals (BC topic)
6.14 Selecting Techniques for Antidifferentiation
Review - Unit 6 Unit 7 - Differential Equations
7.1 Modeling Situations with Differential Equations
7.2 Verifying Solutions for Differential Equations
7.3 Sketching Slope Fields
7.4 Reasoning Using Slope Fields
7.5 Euler's Method (BC topic)
7.6 General Solutions Using Separation of Variables
7.7 Particular Solutions using Initial Conditions and
Separation of Variables
7.8 Exponential Models with Differential Equations
7.9 Logistic Models with Differential Equations (BC topic)
Review - Unit 7 Unit 8 - Applications of Integration
8.1 Average Value of a Function on an Interval
8.2 Position, Velocity, and Acceleration Using Integrals
8.3 Using Accumulation Functions and Definite Integrals
in Applied Contexts
8.4 Area Between Curves (with respect to x)
8.5 Area Between Curves (with respect to y)
8.6 Area Between Curves - More than Two Intersections
Mid-Unit Review - Unit 8
8.7 Cross Sections: Squares and Rectangles
8.8 Cross Sections: Triangles and Semicircles
8.9 Disc Method: Revolving Around the x- or y- Axis
8.10 Disc Method: Revolving Around Other Axes
8.11 Washer Method: Revolving Around the x- or y- Axis
8.12 Washer Method: Revolving Around Other Axes
8.13 The Arc Length of a Smooth, Planar Curve and
Distance Traveled (BC topic)
Review - Unit 8 Unit 9 - Parametric Equations, Polar Coordinates, and Vector-Valued Functions (BC topics)
9.1 Defining and Differentiating Parametric Equations
9.2 Second Derivatives of Parametric Equations
9.3 Arc Lengths of Curves (Parametric Equations)
9.4 Defining and Differentiating Vector-Valued Functions
9.5 Integrating Vector-Valued Functions
9.6 Solving Motion Problems Using Parametric and
Vector-Valued Functions
9.7 Defining Polar Coordinates and Differentiating in
Polar Form
9.8 Find the Area of a Polar Region or the Area Bounded
by a Single Polar Curve
9.9 Finding the Area of the Region Bounded by Two
Polar Curves
Review - Unit 9 Unit 10 - Infinite Sequences and Series (BC topics)
10.1 Defining Convergent and Divergent Infinite Series
10.2 Working with Geometric Series
10.3 The nth Term Test for Divergence
10.4 Integral Test for Convergence
10.5 Harmonic Series and p-Series
10.6 Comparison Tests for Convergence
10.7 Alternating Series Test for Convergence
10.8 Ratio Test for Convergence
10.9 Determining Absolute or Conditional Convergence
10.10 Alternating Series Error Bound
10.11 Finding Taylor Polynomial Approximations of
Functions
10.12 Lagrange Error Bound
10.13 Radius and Interval of Convergence of Power
Series
10.14 Finding Taylor Maclaurin Series for a Function
10.15 Representing Functions as a Power Series
Review - Unit 8 | Version #2
The course below covers all topics for the AP Calculus AB exam, but was built for a 90-minute class that meets every other day.
Lessons and packets are longer because they cover more material.Unit 0 - Calc Prerequisites (Summer Work)
0.1 Things to Know for Calc
0.2 Summer Packet
0.3 Calculator Skillz Unit 1 - Limits
1.1 Limits Graphically
1.2 Limits Analytically
1.3 Asymptotes
1.4 Continuity
Review - Unit 1 Unit 2 - The Derivative
2.1 Average Rate of Change
2.2 Definition of the Derivative
2.3 Differentiability [Calculator Required]
Review - Unit 2 Unit 3 - Basic Differentiation
3.1 Power Rule
3.2 Product and Quotient Rules
3.3 Velocity and other Rates of Change
3.4 Chain Rule
3.5 Trig Derivatives
Review - Unit 3 Unit 4 - More Deriviatvies
4.1 Derivatives of Exp. and Logs
4.2 Inverse Trig Derivatives
4.3 L'Hopital's Rule
Review - Unit 4 Unit 5 - Curve Sketching
5.1 Extrema on an Interval
5.2 First Derivative Test
5.3 Second Derivative Test
Review - Unit 5 Unit 6 - Implicit Differentiation
6.1 Implicit Differentiation
6.2 Related Rates
6.3 Optimization
Review - Unit 6 Unit 7 - Approximation Methods
7.1 Rectangular Approximation Method
7.2 Trapezoidal Approximation Method
Review - Unit 7 Unit 8 - Integration
8.1 Definite Integral
8.2 Fundamental Theorem of Calculus (part 1)
8.3 Antiderivatives (and specific solutions)
Review - Unit 8 Unit 9 - The 2nd Fundamental Theorem of Calculus
9.1 The 2nd FTC
9.2 Trig Integrals
9.3 Average Value (of a function)
9.4 Net Change
Review - Unit 9 Unit 10 - More Integrals
10.1 Slope Fields
10.2 u-Substitution (indefinite integrals)
10.3 u-Substitution (definite integrals)
10.4 Separation of Variables
Review - Unit 10 Unit 11 - Area and Volume
11.1 Area Between Two Curves
11.2 Volume - Disc Method
11.3 Volume - Washer Method
11.4 Perpendicular Cross Sections
Review - Unit 11 |
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When working with tables, the best we can do is estimate the limit value.
Examples
Example 1: Using Tables to Estimate Limits
Use the tables shown below to estimate the value of $$displaystyle lim_{xto 5} f(x)$$.
$$ begin{array}{l|c} {x} & {f(x)} hline 4.5 & 8.32571hline 4.75 & 8.95692hline 4.9 & 8.99084hline 4.99 & 8.99987hline 4.999 & 8.99992hline 4.9999 & 8.99999hline end{array} $$
$$ begin{array}{l|c} {x} & {f(x)} hline 5.5 & 9.64529hline 5.25 & 9.26566hline 5.1 & 9.04215hline 5.01 & 9.00113hline 5.001 & 9.00011hline 5.0001 & 9.00001hline end{array} $$
Step 1Examine what happens as $$x$$ approaches from the left.
As the $$x$$-values approach 5..
$$ begin{array}{l|c} {x} & {f(x)} hline 4.5 & 8.32571hline 4.75 & 8.95692hline 4.9 & 8.99084hline 4.99 & 8.99987hline 4.999 & 8.99992hline 4.9999 & 8.99999hline end{array} $$
..$$f(x)$$ seems to approach 9.
Step 2Examine what happens as $$x$$ approaches from the right.
As the $$x$$-values approach 5..
$$ begin{array}{l|c} {x} & {f(x)} hline 5.5 & 9.64529hline 5.25 & 9.26566hline 5.1 & 9.04215hline 5.01 & 9.00113hline 5.001 & 9.00011hline 5.0001 & 9.00001hline end{array} $$
..$$f(x)$$ seems to approach 9.
Step 3If the function seems to approach the same value from both directions, then that is the estimate of the limit value.
Answer: $$displaystyle lim_{xto 5} f(x) approx 9$$.
Example 2: Using Tables to Estimate Limits
Using the tables below, estimate $$displaystyle lim_{xto-8} f(x)$$.
$$ begin{array}{l|c} {x} & {f(x)} hline -8.5 & 13.1365hline -8.1 & -2.4336hline -8.01 & -2.91313hline -8.001 & -2.99131hline -8.0001 & -2.99913hline -8.00001 & -2.99991hline end{array} $$
$$ begin{array}{l|c} {x} & {f(x)} hline -7.5 & -6hline -7.9 & -5.5hline -7.99 & -5.15hline -7.999 & -5.015hline -7.9999 & -5.0015hline -7.99999 & -5.00015hline end{array} $$
Step 1Examine what happens as $$x$$ approaches from the left.
As the $$x$$-values approach -8..
$$ begin{array}{l|c} {x} & {f(x)} hline -8.5 & 13.1365hline -8.1 & -2.4336hline -8.01 & -2.91313hline -8.001 & -2.99131hline -8.0001 & -2.99913hline -8.00001 & -2.99991hline end{array} $$
..$$f(x)$$ seems to approach -3.
Step 2Examine what happens as $$x$$ approaches from the right.
As the $$x$$-values approach -8..
$$ begin{array}{l|c} {x} & {f(x)} hline -7.5 & -4hline -7.9 & -3.5hline -7.99 & -3.15hline -7.999 & -3.015hline -7.9999 & -3.0015hline -7.99999 & -3.00015hline end{array} $$
..$$f(x)$$ seems to approach -3.
Step 3If the function seems to approach different values, then the limit does not exist.
Answer: $$displaystyle lim_{xto-8} f(x)$$ does not exist.
Practice Problems
Step 1Examine what happens as $$x$$ approaches from the left.
As the $$x$$-values approach 12 from the left..
$$ begin{array}{l|c} {x} & {f(x)} hline 11.5 & 7hline 11.9 & 7.5hline 11.99 & 7.9hline 11.999 & 7.99hline 11.9999 & 7.999hline 11.99999 & 7.9999hline end{array} $$
..$$f(x)$$ seems to approach 8.
Step 2Examine what happens as $$x$$ approaches from the right.
As the $$x$$-values approach 12 from the right
$$ begin{array}{l|c} {x} & {f(x))} hline 12.5 & 8.5hline 12.1 & 8.1hline 12.01 & 8.01hline 12.001 & 8.001hline 12.0001 & 8.0001hline 12.00001 & 8.00001hline end{array} $$
..f(x) seems to approach 8.
Step 3If the function seems to approach the same value from both directions, then that is the estimate of the limit value.
$$displaystyle lim_{xto 12} f(x) approx 8$$.
Step 1League of legends download free for mactreeal. Examine what happens as $$x$$ approaches from the left.
As the $$x$$-values approach $$frac{1}{2}$$ from the left..
$$ begin{array}{l|c} {x} & {f(x)} hline 0 & 1.7hline 0.2 & 1.75hline 0.4 & 1.795hline 0.45 & 1.7995hline 0.49 & 1.79995hline 0.499 & 1.79999hline end{array} $$
..$$f(x)$$ seems to approach 1.8.
Step 2Examine what happens as $$x$$ approaches from the right.
As the $$x$$-values approach $$frac{1}{2}$$ from the right
$$ begin{array}{l|c} {x} & {f(x)} hline 1 & -2.44445hline 0.8 & -2.55556hline 0.6 & -2.66667hline 0.55 & -2.77778hline 0.51 & -2.88889hline 0.501 & -2.99999hline end{array} $$
$$f(x)$$ seems to approach -3.
Step 3If the function seems to approach different values, then the limit does not exist.
$$limlimits_{xtofrac{1}{2}} f(x)$$ does not exist.
Step 1Examine what happens as $$x$$ approaches from the left.
As the $$x$$-values approach 0.75 from the left..
$$ begin{array}{l|c} {x} & {f(x)} hline 0.7 & 0.1hline 0.72 & -0.01hline 0.74 & 0.001hline 0.749 & -0.0001hline 0.7499 & 0.00001hline 0.74999 & -0.000001hline end{array} $$
..$$f(x)$$ seems to approach 0.
Step 2Examine what happens as $$x$$ approaches from the right.
1.8 Limits With Infinityap Calculus Pdf
As the $$x$$-values approach 0.75 from the right..
$$ begin{array}{l|c} {x} & {f(x)} hline 0.8 & 0.3hline 0.78 & -0.06hline 0.76 & 0.0012hline 0.751 & -0.0006hline 0.7501 & 0.00003hline 0.75001 & -0.000006hline end{array} $$
..$$f(x)$$ seems to approach 0.
1.8 Limits With Infinityap Calculus Solutions
Step 3
Step 1Examine what happens as $$x$$ approaches from the left.
As the $$x$$-values approach 10 from the left..
$$ begin{array}{l|c} {x} & {f(x)} hline 9.5 & 2.3hline 9.9 & 1.8hline 9.99 & 8.3hline 9.999 & 0.8hline 9.9999 & 9.8hline 9.99999 & 2.6hline end{array}; $$
..$$f(x)$$ doesn't seem to approach anything.
1.8 Limits With Infinityap Calculus Solution
Step 21.8 Limits With Infinityap Calculus Answers
Examine what happens as $$x$$ approaches from the right
As the $$x$$-values approach 10 from the right..
$$ begin{array}{l|c} {x} & {f(x)} hline 10.5 & 1.1hline 10.1 & 5.8hline 10.01 & 3.6hline 10.001 & 2.9hline 10.0001 & 5.4hline 10.00001 & 12.5hline end{array}; $$
..$$f(x)$$ doesn’t seem to approach anything.
Step 3 The function doesn't seem to approach a particular value, so the limit does not exist.
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Step 1Examine what happens as $$x$$ approaches from the left.
As the $$x$$-values approach -3 from the left..
$$ begin{array}{l|c} {x} & {f(x)} hline -4 & 6hline -3.5 & 61hline -3.1 & 611hline -3.01 & 6111hline -3.001 & 61,111hline -3.0001 & 611,111hline end{array} $$
..$$f(x)$$ keeps getting larger.
Step 2Examine what happens as $$x$$ approaches from the right.
As the $$x$$-values approach 3 from the right..
$$ begin{array}{l|c} {x} & {f(x)} hline -2 & 7hline -2.5 & 72hline -2.9 & 788hline -2.99 & 7656hline -2.999 & 77,701hline -2.9999 & 711,000hline end{array} $$
..$$f(x)$$ keeps getting larger.
Step 3The function doesn't seem to approach a particular value, so the limit does not exist.
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