Problem 16 - APEX Calculus

Question

Numerically approximate the following limits: (a) \(\lim _{x \rightarrow 3^{-}} f(x)\) (b) \(\lim _{x \rightarrow 3^{+}} f(x)\) (c) \(\lim _{x \rightarrow 3} f(x)\) $$ f(x)=\frac{x^{2}+5 x-36}{x^{3}-5 x^{2}+3 x+9} $$

Step-by-Step Solution

Answer

(a) Positive infinity, (b) Negative infinity, (c) Limit does not exist.

Step 1: Understand the function

The function given is \( f(x) = \frac{x^2 + 5x - 36}{x^3 - 5x^2 + 3x + 9} \). To find the limits as \( x \) approaches 3, we need to simplify the expression to understand its behavior near \( x = 3 \).

Step 2: Check if the function is undefined at x = 3

Evaluate both the numerator \( x^2 + 5x - 36 \) and the denominator \( x^3 - 5x^2 + 3x + 9 \) at \( x = 3 \).- For the numerator: \( 3^2 + 5 \cdot 3 - 36 = 9 + 15 - 36 = -12 \) (non-zero).- For the denominator: \( 3^3 - 5 \cdot 3^2 + 3 \cdot 3 + 9 = 27 - 45 + 9 + 9 = 0 \).The denominator is zero, so the function is undefined at \( x = 3 \).

Step 3: Simplify the function

Since the denominator is zero when \( x = 3 \), check for common factors in the numerator and denominator. Use synthetic division for the polynomial:- Divide \( x^3 - 5x^2 + 3x + 9 \) by \( x - 3 \). - Synthetic division confirms that \( x - 3 \) is a factor, resulting in the quotient \( x^2 - 2x - 3 \).- Factor the numerator: \( x^2 + 5x - 36 = (x - 4)(x + 9) \).- The simplified function is \( f(x) = \frac{(x - 4)(x + 9)}{(x - 3)(x^2 - 2x - 3)} \) with a factor of \( x - 3 \) cancelled, rewritten as \( f(x) = \frac{(x - 4)(x + 9)}{(x - 3)(x - 3)} \). Then as \( x \to 3 \), it behaves like \( \frac{(x - 4)(x + 9)}{x - 3} \).

Step 4: Find the left-hand limit (x -> 3^-)

Substitute values slightly less than 3 into the simplified expression \( f(x) \approx \frac{(x - 4)(x + 9)}{x - 3} \). - As \( x \) approaches 3 from the left, \( x - 4 \) is negative, and \( x - 3 \) is negative, making the fraction positive once they cancel.- Compute: using a small number like 2.9, check \( \approx \frac{(-1.1)(11.9)}{-0.1} \), indicating the limit approaches positive infinity.

Step 5: Find the right-hand limit (x -> 3^+)

Substitute values slightly more than 3 into the simplified function. - As \( x \) approaches 3 from the right, \( x - 4 \) is negative, but \( x - 3 \) is positive, making the fraction negative as they cancel.- Compute using a small number like 3.1: evaluate \( \approx \frac{(-0.9)(12.1)}{0.1} \), indicating the limit approaches negative infinity.

Step 6: Determine if the limit exists at x -> 3

Compare both one-sided limits:- The left-hand limit \( \lim _{x \to 3^-} f(x) = \infty \).- The right-hand limit \( \lim _{x \to 3^+} f(x) = -\infty \).- Since the left-hand and right-hand limits are not equal, the limit \( \lim _{x \to 3} f(x) \) does not exist.

Key Concepts

One-Sided Limits

When dealing with one-sided limits, we examine the behavior of a function as it approaches a particular point from either the left or the right.
This is essential when a function is not defined at that exact point, just like in our example, where the function becomes undefined at \(x = 3\).

• The left-hand limit, denoted as \(\lim_{{x \to c^-}} f(x)\), is the value that \(f(x)\) approaches as \(x\) gets closer to a specific point \(c\) from the left (values less than \(c\)).
• The right-hand limit, expressed as \(\lim_{{x \to c^+}} f(x)\), examines the value \(f(x)\) approaches as \(x\) inches closer to \(c\) from the right (values greater than \(c\)).

For our function \(f(x) = \frac{(x - 4)(x + 9)}{x - 3}\), although it's undefined at \(x = 3\), we look at its trend as we test numbers like 2.9 (left side) and 3.1 (right side). This analysis shows that the left limit approaches infinity and the right approaches negative infinity, meaning the overall limit at \(x = 3\) does not exist.
It illustrates how different behaviors on each side of a point can affect the overall limit.

Synthetic Division

Synthetic division is a simplified method of dividing polynomials, especially useful when dividing by linear factors such as \(x - 3\).
It's more streamlined compared to traditional polynomial division.In our example, we applied synthetic division to check if \(x - 3\) is a factor of the denominator \(x^3 - 5x^2 + 3x + 9\). This involves:

• Using the root of the factor, in this case, 3, and writing the coefficients of the polynomial, which are 1, -5, 3, and 9.
• The division process allows us to determine the quotient, \(x^2 - 2x - 3\), upon successfully dividing the polynomial by \(x - 3\).

This method effectively verifies the factorization and helps in simplifying the function \(f(x)\) by confirming that parts of it cancel out, leading to a simpler expression.
Such simplification is crucial for evaluating limits that might otherwise appear complicated.

Polynomial Factorization

Factorization represents a crucial tool in algebra for simplifying expressions, assessing limits, and solving equations. It involves breaking down a polynomial into products of simpler polynomials.
In the context of limits and undefined points, factorization is essential in handling and simplifying complex polynomial expressions.For the numerator \(x^2 + 5x - 36\) in our example, factorization transforms it into the product \((x - 4)(x + 9)\).
This step helps to perform cancellations with the denominator, improving our understanding of the behavior of the function around \(x = 3\).

• The factorization involves finding pairs that multiply to the constant term \(c\) and add up to the linear coefficient \(b\) in \(ax^2 + bx + c\).
• Thus, identifying \(x - 3\) as a factor in both numerator and denominator allows for partial fraction simplification, a common method in calculus to tackle expressions approaching undefined values.

By eliminating common factors, we expose the essence of the function's behavior near points where it appears undefined.
Factorization, in this way, becomes a vital skill in understanding more intricate mathematical concepts such as limits.