Problem 28 - Single Variable Calculus

Question

Find the limit. $$ \lim _{x \rightarrow \infty}\left(1+\frac{1}{x}\right)\left(\frac{x^{2}+1}{x^{2}-1}\right) $$

Step-by-Step Solution

Answer

The limit as x approaches infinity for the given function is 1. To find this, first simplify the expression to a single fraction: \(\frac{(x + 1)(x^2 + 1)}{x(x^2 - 1)}\). Since the highest degrees for both the numerator and denominator are the same (3), and both leading coefficients are 1, the limit is \(\frac{1}{1} = 1\).

Step 1: Identify the Limit

Find the limit as x approaches infinity for the given function: \[ \lim_{x \rightarrow \infty}\left(1 + \frac{1}{x} \right)\left(\frac{x^2 + 1}{x^2 - 1}\right) \]

Step 2: Simplify the expression

First, we need to simplify the expression before applying the limit. To do this, we need to rewrite the expression as a single fraction: \[ \frac{(1 + \frac{1}{x})(x^2 + 1)}{(x^2 - 1)} \]

Step 3: Combine the fractions

Now, we combine the fractions and simplify the expression: \[ \frac{(x + 1)(x^2 + 1)}{x(x^2 - 1)} \]

Step 4: Determine the degree of the numerator and denominator

Determine the highest degree of the polynomial terms in both the numerator and the denominator: - For the numerator: \((x + 1)(x^2 + 1) = x^3 + x + x^2 + 1\), the highest degree is 3. - For the denominator: \(x(x^2 - 1) = x^3 - x\), the highest degree is 3. Since the highest degrees for both numerator and denominator are the same, we can proceed with the next step.

Step 5: Find the leading coefficients of the numerator and denominator

In order to get the limit, we need to find the leading coefficients for both numerator and denominator, as the highest degrees were the same: - For the numerator, the leading coefficient is 1 since it is the coefficient for the \(x^3\) term. - For the denominator, the leading coefficient is 1 since it is the coefficient for the \(x^3\) term as well.

Step 6: Evaluate the limit

Since the degrees are the same, and both leading coefficients are 1, we can evaluate the limit as follows: \[ \lim_{x \rightarrow \infty}\left(1 + \frac{1}{x} \right)\left(\frac{x^2 + 1}{x^2 - 1}\right) = \frac{1}{1} = 1 \] So, the limit as x approaches infinity for the given function is 1.

Key Concepts

Limit Evaluation

When dealing with calculus limits, the primary aim is to determine the behavior of a function as the input approaches a certain point, often infinity. In our exercise, we're interested in how the expression \( \lim_{x \rightarrow \infty}\left(1 + \frac{1}{x} \right)\left(\frac{x^2 + 1}{x^2 - 1}\right) \) behaves as \( x \) becomes very large.
To evaluate this, we take advantage of known calculus strategies such as simplifying complex expressions, which often involve polynomial terms. As \( x \) grows, expressions like \( 1 + \frac{1}{x} \) approach 1 since \( \frac{1}{x} \) tends towards 0.
Another aspect is that certain terms in a rational function (a ratio of polynomials) may dominate at infinity. By focusing on the terms of the highest degree, we can predict the function's end behavior. This simplification helps in making limits manageable and offers insights into the function's growth rate as \( x \rightarrow \infty \).

Polynomial Degrees

In the context of evaluating limits, understanding polynomial degrees is crucial. A polynomial's degree is the highest power of \( x \) in its expression, which gives us valuable information on the function's behavior as \( x \rightarrow \infty \).
In the given problem, we identify the degrees of the polynomials in both the numerator and the denominator:

• In the numerator: \( (x + 1)(x^2 + 1) = x^3 + x^2 + x + 1 \), with a highest degree of 3.
• In the denominator: \( x(x^2 - 1) = x^3 - x \), also with a highest degree of 3.

When both the numerator and the denominator have the same highest degree, the polynomial's terms of the same degree will essentially determine the end behavior of the fraction as \( x \rightarrow \infty \). This is a key consideration when simplifying expressions to evaluate limits.

Leading Coefficients

Leading coefficients are the coefficients of the terms with the highest degree in a polynomial. They play a significant role in determining the limit of a rational function where the numerator and the denominator have the same degree.
In our example, after simplifying the expression to focus on the terms \( x^3 \) in both the numerator and denominator, which are of equal degree, we recognize:

• The leading coefficient of the numerator is 1 (from the term \( x^3 \)).
• The leading coefficient of the denominator is also 1 (from the term \( x^3 \)).

With matching degrees and leading coefficients, the overall limit simplifies straightforwardly to the ratio of these coefficients: \( \frac{1}{1} = 1 \).
By focusing on leading coefficients, evaluating expressions simplifies complex calculations, allowing us to quickly conclude their limits as \( x \rightarrow \infty \). This method becomes particularly useful in calculus when working with large numbers or expressions that seem difficult at first glance.