Question

Numerically approximate the following limits: (a) \(\lim _{x \rightarrow 3^{-}} f(x)\) (b) \(\lim _{x \rightarrow 3^{+}} f(x)\) (c) \(\lim _{x \rightarrow 3} f(x)\) $$ f(x)=\frac{x^{2}-9 x+18}{x^{2}-x-6} $$

Short answer

(a) -0.6, (b) -0.6, (c) -0.6.

Step-by-step solution

Analyze the function

Start by simplifying the function \( f(x) = \frac{x^2 - 9x + 18}{x^2 - x - 6} \). Recognize the quadratic expressions can be factored as \( x^2 - 9x + 18 = (x-3)(x-6) \) and \( x^2 - x - 6 = (x-3)(x+2) \). Thus, the function simplifies to \( f(x) = \frac{(x-3)(x-6)}{(x-3)(x+2)} \) for \( x eq 3 \).

Simplify the function further

Since \((x-3)\) is a common factor in both the numerator and denominator, cancel it out to simplify: \( f(x) = \frac{x-6}{x+2} \) when \( x eq 3 \).

Evaluate the left-hand limit (part a)

For part (a), evaluate \( \lim_{x \to 3^{-}} f(x) \). Since the simplified function is \( \frac{x-6}{x+2} \), plug in numbers slightly less than 3 to evaluate it numerically. As \( x \to 3^{-} \), \( \frac{3-6}{3+2} = \frac{-3}{5} = -0.6 \). Thus, \( \lim_{x \to 3^{-}} f(x) = -0.6 \).

Evaluate the right-hand limit (part b)

For part (b), evaluate \( \lim_{x \to 3^{+}} f(x) \). Similarly, use the simplified function, \( \frac{x-6}{x+2} \). As \( x \to 3^{+} \), \( \frac{3-6}{3+2} = \frac{-3}{5} = -0.6 \). Thus, \( \lim_{x \to 3^{+}} f(x) = -0.6 \).

Evaluate the two-sided limit (part c)

For part (c), evaluate \( \lim_{x \to 3} f(x) \). Since both \( \lim_{x \to 3^{-}} f(x) = -0.6 \) and \( \lim_{x \to 3^{+}} f(x) = -0.6 \), the two-sided limit exists. Therefore, \( \lim_{x \to 3} f(x) = -0.6 \).

Key concepts

Quadratic Expressions

Quadratic expressions are polynomial expressions of degree two. These expressions take the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) represents the variable. Quadratics can be graphed as parabolas and have a range of applications in mathematics.
Understanding quadratic expressions is vital for simplifying functions, particularly when calculating limits or solving equations. The roots of the quadratic expression, or the values of \( x \) that make the expression equal to zero, can be determined using methods like factoring, completing the square, or the quadratic formula.
When dealing with limits, as in the provided exercise, recognizing and using the quadratic expressions given in the numerator and denominator is a key step in simplifying the function before proceeding further with limit calculations.

Factoring

Factoring is the process of breaking down expressions into products of simpler expressions or numbers. It is particularly useful for simplifying quadratic expressions into linear factors, which can then be analyzed or canceled if they appear in both the numerator and denominator of a fraction.
In the provided solution, the quadratic expressions \( x^2 - 9x + 18 \) and \( x^2 - x - 6 \) are factored.

• The expression \( x^2 - 9x + 18 \) factors into \((x-3)(x-6)\),
• while \( x^2 - x - 6 \) factors into \((x-3)(x+2)\).

Factoring revealed a common factor, \( x-3 \), which could be canceled out, allowing the function to simplify to \( \frac{x-6}{x+2} \). By simplifying the expression, the calculation of limits becomes easier, as shown in the original solution. This process demonstrates how factoring can simplify seemingly complex problems into manageable parts.

Left-Hand Limit

Left-hand limits examine the behavior of a function as the input approaches a particular value from the left, or from values smaller than the target.
For example, to approximate \( \lim_{x \to 3^{-}} f(x) \), we need to substitute values of \( x \) that are slightly less than 3 into the function. It's as if we're sneaking up to 3 from the smaller numbers.
In the context of the simplified function \( \frac{x-6}{x+2} \), plugging numbers like 2.9 or 2.99 into \( x \) helps us see how the numbers behave as \( x \) approaches 3 from the left. The function simplifies to values that approach \(-0.6\) indicating the left-hand limit of the function as \( -0.6 \). This understanding aids in gauging the function's behavior close to the critical point.

Right-Hand Limit

Right-hand limits focus on the function's behavior as the input approaches the particular value from the right, or from values larger than the target.
Consider \( \lim_{x \to 3^{+}} f(x) \) for a right-hand limit. This involves substituting values of \( x \) that are slightly more than 3. You'd check what happens when \( x \) approaches 3 from the larger side.
Using the simplified function \( \frac{x-6}{x+2} \), you can substitute numbers like 3.1 or 3.01 to see how the outputs sync as \( x \) nears 3 from the right. The results converge to \(-0.6\), indicating that the right-hand limit is \(-0.6\). Understanding both left and right-hand limits can reveal properties about a function's continuity at that point.