Example (2.2) - Finding the limit of a function #14 (Calc)

Cory Sheeley
17 Oct 201805:24
EducationalLearning
32 Likes 10 Comments

TLDRIn this educational video, the presenter demonstrates how to find the limit of a function as 'u' approaches 3. Initially, plugging in 3 results in a domain error due to division by zero. The solution involves factoring the numerator and denominator, breaking down the expression into simpler components. Through algebraic manipulation, the presenter simplifies the function, cancels out terms, and evaluates the limit, concluding with the result of 4. This process highlights the importance of factoring in solving limits and provides a clear example for students to follow.

Takeaways
  • 📚 The video is an educational example from a math textbook, specifically section 3.2, focusing on finding a limit as a variable approaches a certain value.
  • 🔍 The limit to be found is as 'u' approaches 3 for the function \( A^u \) to the 4th minus 81 over \( u^3 \) minus 27.
  • ❌ Initially, plugging in 3 for 'u' results in a domain error because \( 3^3 \) minus 27 equals zero, making the denominator undefined.
  • 🔑 The solution involves factoring the numerator and denominator to simplify the expression and avoid the domain error.
  • 📐 The numerator \( u^4 - 81 \) is factored into \( (u^2 + 9)(u^2 - 9) \), which further breaks down into \( (u+3)(u-3)(u^2+9) \).
  • 📏 The denominator \( u^3 - 27 \) is factored using the difference of cubes formula, resulting in \( (u-3)(u^2+3u+9) \).
  • 🔄 After factoring, terms \( (u-3) \) in the numerator and denominator cancel each other out, simplifying the expression.
  • 🧩 The simplified expression is \( (u^2 + 9)(u+3) \) over \( u^2 + 3u + 9 \).
  • 🔢 Substituting 3 for 'u' in the simplified expression allows for the evaluation of the limit without any undefined terms.
  • 📈 The final calculation of the limit as 'u' approaches 3 results in the value 4, after simplifying the fraction \( \frac{18 \times 6}{27} \) to \( \frac{108}{27} \) and then further to \( \frac{36}{9} \).
  • 🎯 The key takeaway is that the limit of the given function as 'u' approaches 3 is 4, achieved through careful factoring and simplification.
Q & A

  • What is the mathematical problem being discussed in the script?

    -The script discusses the process of finding the limit as 'u' approaches 3 for the function (u^4 - 81) / (u^3 - 27).

  • Why can't we directly substitute 3 into the function to find the limit?

    -Direct substitution is not possible because when 'u' equals 3, the denominator (u^3 - 27) becomes zero, resulting in a domain error and making the function undefined.

  • What algebraic technique is used to simplify the function before finding the limit?

    -The technique used is factoring. The numerator u^4 - 81 and the denominator u^3 - 27 are both factored to simplify the expression.

  • What is the factored form of u^4 - 81?

    -The factored form of u^4 - 81 is (u^2 + 9)(u^2 - 9), which can further be broken down into (u^2 + 9)(u + 3)(u - 3).

  • How is the denominator u^3 - 27 factored?

    -The denominator is factored using the formula for a^3 - b^3, which results in (u - 3)(u^2 + 3u + 9).

  • What happens when we cancel out the common factor (u - 3) from the numerator and denominator?

    -After canceling out (u - 3), the expression simplifies to (u^2 + 9)(u + 3) / (u^2 + 3u + 9).

  • Why is it important to cancel out the common factor (u - 3)?

    -Canceling out the common factor (u - 3) is important because it removes the undefined part of the function and allows us to evaluate the limit as 'u' approaches 3.

  • What is the simplified form of the function after canceling out (u - 3)?

    -The simplified form of the function is u^2 + 9u + 9 / u^2 + 3u + 9.

  • How do we evaluate the limit as 'u' approaches 3 for the simplified function?

    -By substituting 'u' with 3 in the simplified function and performing the arithmetic, we can find the limit.

  • What is the final result of the limit as 'u' approaches 3 for the given function?

    -The final result of the limit as 'u' approaches 3 for the given function is 4.

  • What is the significance of the limit being 4 in the context of the script?

    -The limit being 4 signifies the value that the function approaches as 'u' gets arbitrarily close to 3, which is a key concept in calculus.

Outlines
00:00
📚 Calculating the Limit of a Polynomial Function

This paragraph discusses the process of finding the limit of a polynomial function as 'u' approaches 3. The function given is 'u^4 - 81' over 'u^3 - 27'. Initially, the presenter tries to substitute 'u' with 3, but encounters a domain error due to the denominator becoming zero. To resolve this, the presenter factors the polynomials, using algebraic identities to simplify the expression. The 'u^4 - 81' is factored into '(u^2 + 9)(u^2 - 9)', and further into '(u + 3)(u - 3)(u^2 + 3u + 9)'. Similarly, 'u^3 - 27' is factored into '(u - 3)(u^2 + 3u + 9)'. After canceling out the common factor of 'u - 3', the function simplifies to 'u^2 + 9' over 'u^2 + 3u + 9'. The presenter then evaluates the limit by substituting 'u' with 3, resulting in a simplified expression that can be easily calculated.

05:01
🔍 Final Evaluation of the Limit

In this paragraph, the presenter concludes the calculation of the limit. After simplifying the function to 'u^2 + 9' over 'u^2 + 3u + 9' and canceling out the common factors, the presenter substitutes 'u' with 3 to find the limit. The resulting expression is '3^2 + 9' over '3^2 + 3*3 + 9'. After performing the arithmetic, the presenter simplifies the fraction to '18*6' over '27', which further simplifies to '108' over '27'. The final step is to divide both the numerator and the denominator by 9, yielding the limit as '4'. The presenter confirms that the limit of the original function as 'u' approaches 3 is indeed 4.

Mindmap
Keywords
💡Limit
In calculus, the 'limit' refers to the value that a function or sequence approaches as the input or index approaches some value. In the video, the limit is being sought as 'u' approaches 3 for a given function. The script discusses the process of finding this limit by evaluating the function at the point where 'u' is 3, which initially leads to a domain error, indicating the need for further mathematical manipulation.
💡Factoring
Factoring is the process of breaking down a polynomial into a product of other polynomials. In the script, the presenter mentions factoring as a method to simplify the expression given in the limit problem. The expression 'u to the 4th minus 81' is factored into 'u squared plus 9 u squared minus 9', which is a key step in solving the limit.
💡Domain Error
A 'domain error' in mathematics occurs when a function is evaluated at a point where it is not defined. In the context of the video, when the presenter plugs in 3 for 'u' in the original expression, it results in a domain error because the denominator becomes zero, which is undefined in real numbers.
💡Algebra 2
Algebra 2 is a high school mathematics course that typically covers more advanced algebraic concepts than Algebra 1. The script refers to Algebra 2 as a background knowledge that might help in understanding how to factor the given expression, suggesting that the concepts used in the video are at an advanced algebraic level.
💡Cube
In mathematics, a 'cube' refers to a number raised to the power of three. The script discusses the process of factoring a difference of cubes, which is a specific algebraic identity used to simplify expressions of the form 'a cubed minus b cubed'.
💡Difference of Cubes
The 'difference of cubes' is an algebraic identity that states that for any two numbers a and b, the expression 'a cubed minus b cubed' can be factored into '(a minus b)(a squared plus ab plus b squared)'. The script uses this identity to factor the denominator of the given expression.
💡Factor Line
A 'factor line' is a method used in algebra to break down expressions, especially for factoring polynomials. The script mentions using a factor line to factor the expression 'u cubed minus 27' into '(u minus 3)(u squared plus 3u plus 9)'.
💡Cancellation
In algebra, 'cancellation' refers to the process of removing common factors from the numerator and denominator of a fraction, simplifying the expression. The script describes the cancellation of 'u minus 3' from both the numerator and denominator, which simplifies the expression and allows for the evaluation of the limit.
💡Evaluation
Evaluation in the context of limits means substituting the value that the variable approaches into the simplified expression to find the limit's value. The script describes the process of evaluating the limit by substituting 3 for 'u' in the simplified expression and performing the necessary arithmetic to find the limit.
💡Undefined
In mathematics, a function is said to be 'undefined' at a certain point if it does not have a value there. The script initially finds the function to be undefined when 'u' is 3 due to division by zero, which is resolved through factoring and cancellation.
💡Simplification
Simplification in mathematics involves making a complex expression easier to understand or work with. The script demonstrates simplification through factoring and cancellation, which ultimately allows for the evaluation of the limit as 'u' approaches 3.
Highlights

Introduction to the problem of finding the limit as u approaches 3 for a given function.

Initial attempt to plug in the value 3 leads to a domain error, indicating the function is undefined at that point.

The necessity of factoring the numerator and denominator to simplify the expression.

Factoring the numerator u^4 - 81 into u^2 + 9u^2 - 9 using algebraic identities.

Explanation of the factoring process for u^4 - 81, highlighting the cancellation of terms.

Factoring the denominator u^3 - 27 using the formula for a^3 - b^3.

Breaking down the denominator into (u - 3)(u^2 + 3u + 9).

Further factoring of u^2 - 9 into (u + 3)(u - 3).

Cancellation of the common factor (u - 3) in the numerator and denominator.

Simplification of the expression to u^2 + 9(u + 3) / (u^2 + 3u + 9).

Evaluation of the limit by substituting u = 3 into the simplified expression.

Calculation of the limit, resulting in a simplified fraction of 18 * 6 over 27.

Further simplification of the fraction to 36 over 9.

Final evaluation of the limit as u approaches 3, which is 4.

Emphasis on the importance of factoring and simplification in evaluating limits.

Demonstration of the algebraic process and the steps taken to solve the limit problem.

Conclusion of the mathematical tutorial with the final answer and a summary of the method used.

Transcripts

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MathematicsLimit CalculationFactoringAlgebraEducationalProblem SolvingU CubeExpression SimplificationDomain ErrorCalculus