Question

Find the following limits or state that they do not exist. Assume \(a, b, c,\) and k are fixed real numbers. $$\lim _{w \rightarrow 3^{-}} \frac{|w-3|}{w^{2}-7 w+12}$$

Short answer

Answer: The left-hand limit of the function as \(w\) approaches \(3\) is \(1\).

Step-by-step solution

1. Identify the type of limit

We are asked to find a left-hand limit, which means we are only concerned with the behavior of the function as \(w\) approaches \(3\) from the left side. This is denoted by the notation \(\lim _{w \rightarrow 3^{-}}\).

2. Simplify the absolute value expression

We need to simplify the absolute value expression \(|w-3|\) for \(w\) approaching \(3\) from the left side. Since \(w\) is less than \(3\), we have \((3-w)\) inside the absolute value. So, we rewrite the absolute value expression as: $$|w-3| = 3 - w,$$ for \(w\) approaching \(3\) from the left side.

3. Substitute the simplified expression into the function

Now we will replace the absolute value expression in the given function with the simplified expression \((3-w)\), and the function becomes: $$\frac{3-w}{w^{2}-7 w+12}.$$

4. Factor the denominator

Now, let's factor the quadratic expression in the denominator: $$w^{2}-7 w+12 = (w-3)(w-4).$$

5. Simplify the function

After factoring the denominator, the function becomes: $$\frac{3-w}{(w-3)(w-4)}.$$ Now, since the numerator \((3-w)\) and the \((w-3)\) in the denominator are opposite, we can cancel out the common factor: $$\frac{3-w}{(w-3)(w-4)} = -\frac{1}{w-4}.$$

6. Evaluate the limit

Now that the function is simplified, we can evaluate the left-hand limit: $$\lim _{w \rightarrow 3^{-}} -\frac{1}{w-4} = -\frac{1}{3-4} = -\frac{1}{-1} = 1.$$ So, the left-hand limit of the function as \(w\) approaches \(3\) is \(1\).

Key concepts

Limits in Calculus

In calculus, the concept of a limit is essential for understanding the behavior of functions as they approach a specific point. Limits can determine the value that a function approaches as the input (or variable) gets infinitely close to a certain number.

In the provided exercise, we're interested in the left-hand limit, which examines what happens to the function as the variable approaches a specific value from the left side on the number line. In notation, this is represented as \( \lim _{w \rightarrow c^{-}} \) for a general value \(c\), and specifically \( \lim _{w \rightarrow 3^{-}} \) for our function. Understanding the left-hand limit is crucial for grasping the overall concept of a limit, which lays the groundwork for derivative and integration—the two central pillars of calculus.

Absolute Value in Limits

The absolute value of a number is its distance from zero on the number line, regardless of direction. When dealing with limits involving absolute values, the direction we approach the number from (left or right) plays a significant role.

Since we are working with a left-hand limit, we know that the variable is approaching from the left which means the values we consider are slightly less than the limit point. For the expression \( |w-3| \) as \( w \) approaches \( 3 \) from the left, we rewrite the absolute value as \( 3 - w \) since \( w \) is less than \( 3 \) in this context. This understanding is imperative when simplifying functions that include absolute values in their expressions.

Factoring Quadratic Expressions

Factoring is a method of rewriting an expression as the product of its factors. Specifically, factoring quadratic expressions involves breaking down the quadratic into a product of two binomials.

In the given problem, we factor the quadratic expression \(w^{2}-7w+12\) to get \( (w-3)(w-4) \). It's like figuring out what two numbers multiply to give the constant term (here, 12) while adding up to the coefficient of the linear term (here, -7). Factoring is a powerful tool as it allows us to simplify expressions, solve quadratic equations, and understand the behavior of polynomial functions.

Simplifying Algebraic Expressions

Simplifying algebraic expressions can make them easier to work with by reducing them to a more manageable form. This often involves canceling out common factors between the numerator and the denominator of a fraction.

In the exercise solution, after factoring, we identified \(3-w\) and \(w-3\) as opposites, allowing us to cancel the \(w-3\) factor in the numerator and denominator. The simplification step is crucial; it transformed the function into a simple fraction \( -\frac{1}{w-4} \) which we could easily evaluate to find the limit. Simplification often paves the way for further analysis and helps in solving more complex algebraic problems.