AP Precalculus Section 2.3 Example: Limit of an Exponential Function

Random AP Precalculus Problems (I found on the Internet). These are not official AP Collegeboard examples, but they will definitely get the job done!

To find the limit as \(x\) approaches infinity in an exponential equation of the form \(Y = a \cdot b^x\), where \(b is greater than 1\), you can observe the behavior of the exponential term. Here are the steps:

1. **Examine the Growth Factor \(b\):**
- Confirm that \(b\) is greater than 1. This is crucial for exponential growth.

2. **Analyze the Limit:**
- As \(x\) approaches infinity, the exponential term \(b^x\) will increase without bound, assuming \(b is greater than 1\).

3. **Interpret the Limit:**
- The limit as \(x\) approaches infinity will be either positive infinity or undefined, depending on the specific values of \(a\) and \(b\) in your equation.

For example, if your equation is \(Y = 2 \cdot 3^x\), as \(x\) approaches infinity, \(3^x\) grows without bound, and the limit will be positive infinity.

If your equation is \(Y = 0.5 \cdot 2^x\), as \(x\) approaches infinity, \(2^x\) grows without bound, and the limit will be positive infinity multiplied by 0.5, resulting in positive infinity.

In summary, when dealing with exponential growth (\(b is greater 1\)), the limit as \(x\) approaches infinity tends to positive infinity.

The Topics covered in AP Precalculus are...

1.1 Change in Tandem
1.2 Rates of Change
1.3 Rates of Change in Linear and Quadratic Functions
1.4 Polynomial Functions and Rates of Change
1.5 Polynomial Functions and Complex Zeros
1.6 Polynomial Functions and End Behavior
1.7 Rational Functions and End Behavior
1.8 Rational Functions and Zeros
1.9 Rational Functions and Vertical Asymptotes
1.10 Rational Functions and Holes
1.11 Equivalent Representations of Polynomial and Rational Expressions
1.12 Transformations of Functions
1.13 Function Model Selection and Assumption Articulation
1.14 Function Model Construction and Application
2.1 Change in Arithmetic and Geometric Sequences
2.2 Change in Linear and Exponential Functions
2.3 Exponential Functions
2.4 Exponential Function Manipulation
2.5 Exponential Function Context and Data Modeling
2.6 Competing Function Model Validation
2.7 Composition of Functions
2.8 Inverse Functions
2.9 Logarithmic Expressions
2.10 Inverses of Exponential Functions
2.11 Logarithmic Functions
2.12 Logarithmic Function Manipulation
2.13 Exponential and Logarithmic Equations and Inequalities
2.14 Logarithmic Function Context and Data Modeling
2.15 Semi-log Plots
3.1 Periodic Phenomena
3.2 Sine, Cosine, and Tangent
3.3 Sine and Cosine Function Values
3.4 Sine and Cosine Function Graphs
3.5 Sinusoidal Functions
3.6 Sinusoidal Function Transformations
3.7 Sinusoidal Function Context and Data Modeling
3.8 The Tangent Function
3.9 Inverse Trigonometric Functions
3.10 Trigonometric Equations and Inequalities
3.11 The Secant, Cosecant, and Cotangent Functions
3.12 Equivalent Representations of Trigonometric Functions
3.13 Trigonometry and Polar Coordinates
3.14 Polar Function Graphs
3.15 Rates of Change in Polar Functions

I have many informative videos for Pre-Algebra, Algebra 1, Algebra 2, Geometry, Pre-Calculus, and Calculus. Please check it out:

/ nickperich

Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa

#APPrecalculus
#PreCalcProblems
#MathMinds
#AdvancedPreCalc
#TrigTales
#PrecalcPuzzles
#FunctionFiesta
#GraphGoals
#CalcReady
#PreCalcLife
#AlgebraicAdventures
#DerivativeDreams
#IntegrationInsights
#MathematicsMagic
#PreCalcReview
#PrecalcConcepts
#LogarithmLove
#TrigonometryTips
#MathMastermind
#APCalcPrep
#Mathematics
#MathMinds
#Math
#Maths #math #shorts #funny #help #onlineclasses #onlinelearning #online #study