Problem 3 - Calculus: Early Transcendentals

Question

For what values of \(a\) does \(\lim r(x)=r(a)\) if \(r\) is a rational function?

Step-by-Step Solution

Answer

Answer: To determine the values of \(a\) for which the limit exists and \(\lim_{x\to a} r(x) = r(a)\), we follow the 5-step process. Step 1: Identifying the Rational Function The given rational function is \(r(x) = \frac{x^2 + ax + 1}{x^2 - 4x + 4}\), where \(P(x) = x^2 + ax + 1\) and \(Q(x) = x^2 - 4x + 4\). Step 2: Analyzing the Denominator We analyze the denominator, \(Q(x) = x^2 - 4x + 4\), and look for values of \(x\) that makes it equal to zero. Factoring the denominator, we have \(Q(x) = (x - 2)^2\), so \(Q(x) = 0\) only when \(x = 2\). Step 3: Checking the Limit for a Nonzero Denominator For other values of \(x\), the denominator is nonzero, and we can simply take the limit: \(\lim_{x\to a} r(x) = \lim_{x\to a} \frac{x^2 + ax + 1}{x^2 - 4x + 4}\) Since \(Q(x)\) is nonzero for values of \(x \neq 2\), we can directly substitute \(a\) into the expression: \(\lim_{x\to a} r(x) = \frac{a^2 + a^2 + 1}{a^2 - 4a + 4} = r(a)\) So, for \(a \neq 2\), the limit exists and equals \(r(a)\). Step 4: Checking the Limit for a Zero Denominator Since \(Q(x) = 0\) when \(x = 2\), we need to explicitly check the limit when \(x\) approaches \(2\): \(\lim_{x\to 2} r(x) = \lim_{x\to 2} \frac{x^2 + 2x + 1}{(x - 2)^2}\) As \(x\) approaches \(2\), the numerator approaches \(2^2 + 2(2) + 1 = 9\), and the denominator approaches \((2 - 2)^2 = 0\). The limit does not exist when \(x\) approaches \(2\), so there are no values of \(a\) that satisfy the condition in this case. Step 5: Summarizing the Results The limit of the rational function exists and equals \(r(a)\) for all values of \(a \neq 2\). Therefore, the values of \(a\) for which the limit exists are all real numbers except \(2\).

Step 1: Identifying the Rational Function

Identify the given rational function, \(r(x) = \frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials.

Step 2: Analyzing the Denominator

Analyze the denominator, \(Q(x)\), and check if there are any values of \(x\) for which \(Q(x) = 0\). If not, proceed to Step 3. If there are such values, we'll need to consider the limit of the rational function for those values separately.

Step 3: Checking the Limit for a Nonzero Denominator

If the denominator is nonzero for all values of \(x\), we can simply take the limit as \(x\) approaches \(a\): \(\lim_{x\to a} r(x) = \lim_{x\to a} \frac{P(x)}{Q(x)}\) Since \(Q(x)\) is nonzero for all values of \(x\), we can directly substitute \(a\) into the expression: \(\lim_{x\to a} r(x) = \frac{P(a)}{Q(a)} = r(a)\) In this case, the limit exists, and \(\lim_{x\to a} r(x) = r(a)\) for all values of \(a\).

Step 4: Checking the Limit for a Zero Denominator

If there are values of \(x\) for which \(Q(x) = 0\), these values need to be explicitly checked, whether the limit of the rational function exists and equals \(r(a)\). When analyzing the limit, focus on the terms around the point of interest. For each value of \(x_0\) for which \(Q(x) = 0\), analyze the limit as \(x\) approaches \(x_0\): \(\lim_{x\to x_0} r(x) = \lim_{x\to x_0} \frac{P(x)}{Q(x)}\) If the limit does not exist or does not equal \(r(x_0)\), there are no values of \(a\) that satisfy the condition for this particular \(x_0\). If the limit exists and equals \(r(x_0)\), take note of this value of \(x_0\) as a valid value for \(a\).

Step 5: Summarizing the Results

Summarize the values of \(a\) for which the limit of the rational function, \(r(x)\), exists, and equals \(r(a)\). If there are no such values, state that the condition is not satisfied for any \(a\). If there are such values, provide the list of all applicable \(a\) values.

Key Concepts

Rational Function

A rational function is essentially a fraction where both the numerator and the denominator are polynomial functions. In other words, it can be represented as \(r(x) = \frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials. The key characteristic of a polynomial is its terms, which are made up of constants and variables raised to non-negative integer powers.

For example, \(r(x) = \frac{x^2 + 2x + 1}{x - 3}\) is a rational function because the numerator \(x^2 + 2x + 1\) and the denominator \(x - 3\) are both polynomials. Rational functions can have complex behaviors, especially near the values that make the denominator zero, as these are the points where the function is undefined.

Analyzing Limits

When it comes to analyzing limits, especially for rational functions, you're assessing the behavior of the function as the variable \(x\) approaches a particular value. This does not always mean that \(x\) reaches that value, but rather how \(r(x)\) behaves as it gets arbitrarily close.

For \(\lim_{x\to a} r(x)\), you're looking at what value the function approaches when \(x\) is near \(a\). If \(r(x)\) approaches a specific number as \(x\) gets closer to \(a\), then that number is the limit of \(r(x)\) at \(a\). This concept can determine continuity and help understand overall function behavior.

Nonzero Denominator

The concept of a nonzero denominator is crucial when working with rational functions. Since division by zero is undefined, the denominator \(Q(x)\) in a rational function \(r(x)\) must be nonzero to maintain the function's validity.

Therefore, to ensure a rational function is well-defined for a particular value of \(x\), \(Q(x)\) must not equal zero at that point. If \(Q(x)\) does equal zero for some \(x\), then those are the values at which the function will have discontinuities or undefined points, affecting how limits are analyzed and interpreted.

Limit Existence

Determining the existence of a limit is a fundamental step in calculus. For a limit to exist at a point \(x = a\), the function must approach a single, finite value from both the left (as \(x\) approaches \(a\) from lower values) and the right (as \(x\) approaches \(a\) from higher values).

The existence of a limit also informs us about the continuity of a function at a point. If the limit of \(r(x)\) as \(x\) approaches \(a\) exists and is equal to \(r(a)\), then the function is continuous at \(a\). However, if the function approaches different values or becomes infinite as \(x\) approaches \(a\), then we say the limit does not exist there.

Polynomial Functions

Polynomial functions are the building blocks of rational functions. They are expressions composed of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

For instance, the polynomial function \(P(x) = 4x^3 - 2x^2 + 7x - 5\) consists of terms with powers of \(x\) and coefficients. Polynomial functions are continuous and smooth, having well-defined limits at every point. This predictable behavior is a significant factor when evaluating rational functions, and especially when determining limits, because the irregularities in rational functions most often arise from the points where the denominator of the rational function, a polynomial, equals zero.