Limits

The Organic Chemistry Tutor
20 Dec 201906:45
EducationalLearning
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TLDRThe video script demonstrates a method to evaluate a limit involving radicals and fractions. It guides the viewer through the process of simplifying the expression by eliminating the complex fraction and the radicals using multiplication and conjugation techniques. The final step involves direct substitution to find the limit as x approaches 9, resulting in an answer of negative one over fifty-four. The script also suggests verifying the answer by plugging in values close to 9. The detailed explanation helps viewers understand how to tackle similar problems.

Takeaways
  • ๐Ÿ“ The problem involves evaluating a limit as x approaches 9 for a function with radicals and fractions.
  • ๐Ÿšซ Direct substitution is not effective as it results in an indeterminate form (zero over zero).
  • ๐Ÿง  To solve the complex fraction, multiply the numerator and denominator by the common denominator to eliminate fractions.
  • ๐ŸŒŸ After eliminating fractions, the presence of radicals is addressed by multiplying both the numerator and denominator by the conjugate of the radical expression.
  • ๐Ÿ“ˆ The conjugate is found by changing the sign of the radical part in the denominator, thus enabling the cancellation of radical terms.
  • ๐Ÿ”ข By simplifying the expression, we can factor out a negative 1, which transforms the expression into a more manageable form.
  • ๐ŸŽฏ Once the radical terms are gone, the limit can be evaluated by substituting the value of x (9 in this case) into the simplified function.
  • ๐Ÿ” The result is -1/54, which can be verified by checking the function's behavior as x gets closer to 9 from both sides.
  • ๐Ÿ“Š As x approaches 9, the function's value trends towards -0.0185, confirming the correctness of the calculated limit.
  • ๐Ÿ‘ This method demonstrates a systematic approach to evaluating limits with fractions and radicals, which is crucial for understanding more complex mathematical concepts.
Q & A

  • What is the limit being evaluated in the script?

    -The limit being evaluated is the limit as x approaches nine for the function one over the square root of x, minus one over three divided by x minus nine.

  • Why doesn't direct substitution work for this problem?

    -Direct substitution doesn't work because when you plug in 9 into the function, you get an indeterminate form of zero over zero, which does not provide a clear answer.

  • What is the first step to take when dealing with complex fractions in limits?

    -The first step is to eliminate the fractions by multiplying the numerator and the denominator of the main fraction by the least common denominator of the fractions within it.

  • How do you eliminate the radicals in the expression after simplifying the fractions?

    -You eliminate the radicals by multiplying both the numerator and the denominator by the conjugate of the radical expression, which helps in canceling out the radicals.

  • What is the conjugate of the expression involving square root x?

    -The conjugate of the expression involving square root x is 3 minus square root x, with the negative sign changed to a positive sign.

  • What happens when you multiply the numerator and denominator by the conjugate of the expression?

    -Multiplying by the conjugate results in the cancellation of the radical terms, leaving behind a simplified expression without radicals that can be evaluated more easily.

  • How can you check the calculated limit by substituting other values close to 9?

    -By substituting values like 9.1 or 8.99 into the function and observing the results, you can confirm that the function approaches the calculated limit as x gets closer to 9.

  • What is the final answer for the limit as x approaches 9?

    -The final answer is negative one over fifty-four, or as a decimal, approximately -0.0185 repeating.

  • How does the process of evaluating this limit help in understanding complex limit problems?

    -This process demonstrates the techniques of handling fractions and radicals within limit problems, providing a clear method for evaluating limits that may otherwise seem complicated.

  • What is the significance of checking the limit from the left side by substituting a value like 8.99?

    -Checking from the left side ensures that the limit is approached consistently from all directions, confirming the validity of the calculated limit.

  • What is the main takeaway from this script for solving similar problems?

    -The main takeaway is the step-by-step process of handling complex fractions and radicals in limit problems, which includes simplifying expressions, using conjugates, and checking the limit with nearby values.

Outlines
00:00
๐Ÿ“š Evaluating Limits with Fractions and Radicals

This paragraph discusses the process of evaluating a limit as x approaches 9 for a function involving fractions and radicals. Initially, direct substitution is attempted but fails due to the zero over zero indeterminate form. The speaker then suggests multiplying the numerator and denominator by the common denominator to simplify the complex fraction. After simplifying, the expression still contains radicals, so the speaker recommends multiplying both the numerator and the denominator by the conjugate of the radical expression to eliminate the radicals. The process results in a simplified expression, and by substituting x with 9, the limit is found to be negative one over fifty-four (-1/54). The solution is verified by plugging in values close to 9 and confirming the limit.

05:00
๐Ÿ”ข Decimal Representation and Verification of the Limit

In this paragraph, the speaker provides the decimal representation of the limit obtained in the previous section, which is negative 0.0185 repeating. To further verify the solution, the speaker suggests plugging in values such as 9.1 and 9.01 into the original function and observing the results. These values yield -0.01836656 and -0.01, respectively, which approach the limit as x gets closer to 9. Additionally, the speaker checks the limit from the left side by substituting 8.99 and obtaining -0.018534, reinforcing the correctness of the calculated limit. The paragraph concludes with a recap of the method for evaluating limits with fractions and radicals, and the speaker thanks the viewers for watching.

Mindmap
Keywords
๐Ÿ’กlimit
In the context of the video, 'limit' refers to the value that a function approaches as the input (x) gets arbitrarily close to a certain point (in this case, 9). It is a fundamental concept in calculus, used to describe the behavior of functions at specific points or as the input grows without bound. The video demonstrates how to evaluate the limit of a complex function involving radicals and fractions as x approaches 9.
๐Ÿ’กdirect substitution
Direct substitution is a method used in calculus where one simply plugs the value of the input (x) into the function to find the output. In the video, it is mentioned that direct substitution does not work for this particular problem because it results in an indeterminate form (0/0) when x equals 9.
๐Ÿ’กcomplex fraction
A complex fraction is a fraction that has another fraction within its numerator or denominator. In the video, the given function is an example of a complex fraction, which necessitates the use of techniques to simplify the expression before evaluating the limit.
๐Ÿ’กcommon denominator
A common denominator is a number that is shared by the denominators of two or more fractions. In the video, the process of finding the common denominator is essential for simplifying the complex fraction and eliminating the fractions by multiplying the numerator and denominator of the main fraction by this common denominator.
๐Ÿ’กFOIL
FOIL is a mnemonic used to represent the process of expanding the product of two binomials (two expressions in parentheses that are being multiplied). It stands for First, Outer, Inner, Last, which refers to the four terms that result from multiplying two binomials. In the video, the FOIL method is used to multiply the numerator and the denominator after simplifying the complex fraction to get rid of the radicals.
๐Ÿ’กconjugate
In the context of the video, a conjugate refers to a binomial expression where the sign of one term is reversed. Conjugates are used to eliminate radicals by multiplying the numerator and the denominator of a fraction, which helps in simplifying expressions involving square roots.
๐Ÿ’กfactoring
Factoring is the process of breaking down a polynomial into its constituent factors, which are expressions that multiply together to give the original polynomial. In the video, factoring is used to simplify the expression further after applying the conjugate method, allowing for the cancellation of terms.
๐Ÿ’กindeterminate form
An indeterminate form is a type of algebraic expression that does not have a determinate value, often because the expression evaluates to 0/0 or โˆž/โˆž. In the video, the initial attempt at direct substitution results in an indeterminate form, which cannot be used to evaluate the limit.
๐Ÿ’กsimplifying expressions
Simplifying expressions involves using mathematical rules and techniques to reduce complex expressions to their simplest form. In the video, several methods are used to simplify the given function before evaluating the limit, including dealing with complex fractions, finding a common denominator, and using conjugates.
๐Ÿ’กcalculus
Calculus is a branch of mathematics that deals with rates of change and accumulation. It is used to study the behavior of functions, particularly at points of interest such as limits, derivatives, and integrals. The video's main theme revolves around evaluating limits, which is a fundamental concept in calculus.
๐Ÿ’กdecimal approximation
Decimal approximation is the process of expressing a number, especially a fraction, as a decimal for easier understanding or computation. In the video, the limit is not only expressed as a fraction (-1/54) but also as a decimal approximation (-0.0185 repeating) to demonstrate the value the function approaches as x gets closer to 9.
Highlights

The problem involves evaluating a limit as x approaches 9 for a function with radicals and fractions.

Direct substitution is not effective as it results in an indeterminate form of 0/0.

To eliminate the complex fraction, multiply the numerator and denominator by the common denominator of the fractions within the function.

The common denominator is found by multiplying the factors of the denominator, in this case, 3 and โˆšx.

After eliminating the fractions, the function simplifies to a form with radicals only.

To eliminate the radicals, multiply both the numerator and the denominator by the conjugate of the radical expression.

The conjugate of the expression is found by changing the sign of the radical part.

After applying the FOIL method (First, Outer, Inner, Last), the terms in the numerator and denominator start to cancel out.

The simplified function allows for factoring out a negative 1, which further simplifies the expression.

By canceling out the (x - 9) term, the limit can be evaluated without any radicals in the expression.

Substituting x with 9 in the simplified expression gives the limit value.

The final answer for the limit is -1/54, which can be expressed as a decimal, -0.0185 repeating.

To verify the answer, plug in values close to 9, such as 9.1 and 9.01, and observe the function's behavior.

Checking from the left side by plugging in 8.99 also confirms the correctness of the limit value.

This method demonstrates how to evaluate limits with fractions and radicals, providing a clear step-by-step approach.

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