Chapter 2: Problem 15
, find the indicated limit. In most cases, it will be wise to do some algebra first. $$ \lim _{x \rightarrow 3} \frac{x^{4}-18 x^{2}+81}{(x-3)^{2}} $$
Short Answer
Expert verified
The limit is 36.
Step by step solution
01
Identify Indeterminate Form
First, substitute the value of 3 into the function to check if it results in an indeterminate form. Evaluating at \( x = 3 \) gives \( \frac{3^4 - 18 \cdot 3^2 + 81}{(3-3)^2} = \frac{81 - 162 + 81}{0} = \frac{0}{0} \). Thus, the limit is indeterminate, and further manipulation is necessary.
02
Factor the Numerator
The numerator \( x^4 - 18x^2 + 81 \) can be factored as a quadratic in \( x^2 \). Notice that \( x^4 - 18x^2 + 81 = (x^2 - 9)^2 \). This simplifies the numerator from \( x^4 - 18x^2 + 81 \) to \( (x^2 - 9)^2 \).
03
Rewrite using Difference of Squares
Recognize \( x^2 - 9 \) in the factorization as a difference of squares: \( x^2 - 9 = (x - 3)(x + 3) \). So, \( (x^2 - 9)^2 = [(x - 3)(x + 3)]^2 = (x-3)^2(x+3)^2 \).
04
Simplify the Expression
Substitute back into the limit expression: \( \frac{(x-3)^2(x+3)^2}{(x-3)^2} \). The \((x-3)^2\) in the numerator and denominator cancel each other, leaving \( \lim_{x \to 3} (x+3)^2 \).
05
Apply Limit Laws
Now substitute \( x = 3 \) into the simplified expression: \( (x + 3)^2 = (3 + 3)^2 = 6^2 = 36 \). Therefore, the limit is 36.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Indeterminate Forms
In calculus, limits can sometimes result in expressions like \( \frac{0}{0} \) which are called indeterminate forms. These forms don't provide enough information to determine the value of a limit directly. When you encounter an indeterminate form, this signals that more steps are needed, such as algebraic manipulation, to resolve the uncertainty.
Let's think of the function \( \lim _{x \rightarrow 3} \frac{x^{4}-18 x^{2}+81}{(x-3)^{2}} \). When substituting \( x = 3 \), the expression becomes \( \frac{0}{0} \), revealing its indeterminate nature. This outcome tells us it's not possible to determine the limit value without further processing.
To address indeterminate forms, you'll often use algebraic techniques like factoring or finding a common factor to simplify the expression in a way that eliminates these forms and allows limit calculations to proceed.
Let's think of the function \( \lim _{x \rightarrow 3} \frac{x^{4}-18 x^{2}+81}{(x-3)^{2}} \). When substituting \( x = 3 \), the expression becomes \( \frac{0}{0} \), revealing its indeterminate nature. This outcome tells us it's not possible to determine the limit value without further processing.
To address indeterminate forms, you'll often use algebraic techniques like factoring or finding a common factor to simplify the expression in a way that eliminates these forms and allows limit calculations to proceed.
Algebraic Manipulation
Algebraic manipulation involves rearranging mathematical expressions to make them more manageable or to reveal further structure. In the context of limits, algebraic manipulation is crucial for simplifying indeterminate forms so that you can evaluate limits effectively.
In our example, \( x^4 - 18x^2 + 81 \) needs to be simplified in order for us to work around the \( \frac{0}{0} \) indeterminate form. By viewing \( x^4 - 18x^2 + 81 \) as a quadratic in terms of \( (x^2) \), you can factor it more easily to \( (x^2 - 9)^2 \). This step is key to manipulating the expression toward a more useful form.
Algebraic manipulation is the toolset that helps you transform complex algebraic expressions into forms where limits can be calculated effectively by revealing underlying structures you can work with.
In our example, \( x^4 - 18x^2 + 81 \) needs to be simplified in order for us to work around the \( \frac{0}{0} \) indeterminate form. By viewing \( x^4 - 18x^2 + 81 \) as a quadratic in terms of \( (x^2) \), you can factor it more easily to \( (x^2 - 9)^2 \). This step is key to manipulating the expression toward a more useful form.
Algebraic manipulation is the toolset that helps you transform complex algebraic expressions into forms where limits can be calculated effectively by revealing underlying structures you can work with.
Factoring Techniques
Factoring is a fundamental algebraic technique used to simplify expressions and solve equations. In terms of limits, factoring helps to eliminate indeterminate forms by breaking down expressions into products of simpler factors.
Consider the expression \( x^4 - 18x^2 + 81 \) from the exercise. By recognizing this as a perfect square, it can be factored as \( (x^2 - 9)^2 \). Such recognition often requires identifying familiar patterns, such as squares or cubes, that can be expressed in a simpler multiplied format.
Factoring this expression reveals it as \( (x - 3)^2(x + 3)^2 \) in the next step. This transformation is necessary to make other manipulations, such as cancellation, possible, thus resolving the indeterminate nature of the original limit request.
Consider the expression \( x^4 - 18x^2 + 81 \) from the exercise. By recognizing this as a perfect square, it can be factored as \( (x^2 - 9)^2 \). Such recognition often requires identifying familiar patterns, such as squares or cubes, that can be expressed in a simpler multiplied format.
Factoring this expression reveals it as \( (x - 3)^2(x + 3)^2 \) in the next step. This transformation is necessary to make other manipulations, such as cancellation, possible, thus resolving the indeterminate nature of the original limit request.
Difference of Squares
The difference of squares is a particular factoring technique used to simplify expressions that can be written as \( a^2 - b^2 \), which factors into \( (a - b)(a + b) \). This method is especially useful in calculus when dealing with polynomial expressions.
In our example, the term \( x^2 - 9 \) can be seen as a difference of squares since it equals \( (x - 3)(x + 3) \). Recognizing this factorization pattern allows us to further simplify \( (x^2 - 9)^2 \) to \( (x-3)^2(x+3)^2 \).
Using the difference of squares in this limit problem ultimately allows us to cancel out similar terms in the numerator and the denominator, which leads to the resolution of the indeterminate form and the successful calculation of the limit. This illustrates the power and necessity of being able to identify and use such algebraic identities efficiently in calculus.
In our example, the term \( x^2 - 9 \) can be seen as a difference of squares since it equals \( (x - 3)(x + 3) \). Recognizing this factorization pattern allows us to further simplify \( (x^2 - 9)^2 \) to \( (x-3)^2(x+3)^2 \).
Using the difference of squares in this limit problem ultimately allows us to cancel out similar terms in the numerator and the denominator, which leads to the resolution of the indeterminate form and the successful calculation of the limit. This illustrates the power and necessity of being able to identify and use such algebraic identities efficiently in calculus.
