Question

Find the limit of each of the following sequences, using L'Hôpital's rule when appropriate. $$ \frac{(n-1)^{2}}{(n+1)^{2}} $$

Short answer

The limit of the sequence is 1.

Step-by-step solution

Identify the Sequence Limit

We need to find the limit of \( \frac{(n-1)^2}{(n+1)^2} \) as \( n \) approaches infinity. Observe that both the numerator and the denominator are polynomial expressions of degree two, which suggests using asymptotic expansions first before considering L'Hôpital's rule.

Simplify the Expression

Rewrite the expression by expanding both the numerator and the denominator: \[(n-1)^2 = n^2 - 2n + 1\]\[(n+1)^2 = n^2 + 2n + 1\]The sequence becomes \[\frac{n^2 - 2n + 1}{n^2 + 2n + 1}.\]

Factor Out the Highest Power of \( n \)

Divide both the numerator and the denominator by \( n^2 \), which is the highest power of \( n \) in the expression:\[\frac{n^2(1 - \frac{2}{n} + \frac{1}{n^2})}{n^2(1 + \frac{2}{n} + \frac{1}{n^2})} = \frac{1 - \frac{2}{n} + \frac{1}{n^2}}{1 + \frac{2}{n} + \frac{1}{n^2}}\]

Evaluate the Limit as \( n \to \infty \)

As \( n \) approaches infinity, the terms \( \frac{2}{n} \) and \( \frac{1}{n^2} \) become negligible. The expression simplifies to:\[\lim_{{n \to \infty}} \frac{1 - \frac{2}{n} + \frac{1}{n^2}}{1 + \frac{2}{n} + \frac{1}{n^2}} = \frac{1}{1} = 1\]

Confirm with L'Hôpital's Rule (if applicable)

Since both the numerator and denominator are polynomials and their highest powers equate for both, using L'Hôpital's rule is not strictly necessary. However, the algebraic simplification provides a valid confirmation.

Key concepts

Polynomial Expressions

Polynomial expressions are mathematical expressions that involve variables raised to whole number powers, combined using addition or subtraction. They are typically written in the standard form, where the terms are ordered from highest degree to lowest degree. For example, in the sequence given by \[ \frac{(n-1)^2}{(n+1)^2} \] both the numerator and the denominator are polynomial expressions of degree two:

• Numerator: \( (n-1)^2 = n^2 - 2n + 1 \)
• Denominator: \( (n+1)^2 = n^2 + 2n + 1 \)

The importance of polynomial expressions lies in their regular structure, which often allows for straightforward manipulation and simplification. When analyzing limits, recognizing and working with polynomials helps us discern the behavior of sequences and functions as they approach infinity or any other point.

L'Hôpital's Rule

L'Hôpital's Rule is a useful tool for finding limits of indeterminate forms, which occur when both the numerator and the denominator of a fraction approach values that result in an undefined expression, like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). This rule allows us to differentiate the numerator and the denominator separately to find a limit:
When applicable, L'Hôpital's Rule states that:\[ \lim_{{x \to c}} \frac{f(x)}{g(x)} = \lim_{{x \to c}} \frac{f'(x)}{g'(x)} \]whenever the conditions of the rule are met. In our exercise, although L'Hôpital's Rule can be considered, it is not strictly necessary due to asymptotic simplification. However, L'Hôpital's robust approach can still be informative and serve as a check when other methods leave room for doubt. In this example, due to both the numerator and the denominator being of the same degree, careful polynomial simplification already reveals that the sequence's behavior at infinity does not lead to an indeterminate form. It is always good practice to check if simplifications can be made before deciding to apply L'Hôpital's Rule, as this can save time and effort. By understanding and identifying when to use this rule, as well as when algebraic manipulation is simpler, students can tackle a wider range of limit problems with confidence.

Asymptotic Analysis

Asymptotic analysis involves understanding the behavior of functions as they go towards a particular limit, typically infinity. It is commonly used in mathematics and computer science to predict or describe the growth behavior of sequences or algorithms. In limit problems, such as finding the limit of a sequence, asymptotic analysis helps in identifying the dominant terms that influence the outcome:
In the expression \[ \frac{n^2 - 2n + 1}{n^2 + 2n + 1} \]performing asymptotic analysis involves simplifying it by factoring and dividing by highest terms. Here, polynomial expressions behave predictably: terms with lower degrees become negligible as compared to terms with higher degrees as \( n \to \infty \).This simplification leads us to:

• Focus on the \( n^2 \) terms in both the numerator and denominator.
• Recognize that lesser terms \( -2n, +1 \) eventually become negligible.

Asymptotic analysis reveals the limit as simply the ratio of dominant terms:
\( \lim_{{n \to \infty}} \frac{n^2}{n^2} = 1 \).By concentrating on the dominant behavior of the expression, students can reach conclusions faster and solve problems with greater clarity. Understanding the concept helps to solidify reasoning skills in mathematical analysis and sequence behaviors.