Problem 10 - Calculus with Concepts in Calculus

Question

Use the results of this section to evaluate the given limit. $$ \lim _{x \rightarrow 0} f(x), \text { where } f(x)=\left\\{\begin{array}{ll} -\frac{3}{2} x+\frac{1}{4} & \text { for } x \neq \frac{1}{2} \\ 0 & \text { for } x=0 \end{array}\right. $$

Step-by-Step Solution

Answer

The limit is \(\frac{1}{4}\).

Step 1: Check the definition

First, examine the definition of the piecewise function \( f(x) \). The function is defined as \(-\frac{3}{2} x + \frac{1}{4} \) for all \(x eq \frac{1}{2}\) and is specifically 0 when \(x = 0\). Since we are interested in \( \lim _{x \rightarrow 0} f(x) \), we only need the part of the function defined for \(x eq \frac{1}{2}\).

Step 2: Substitute into the relevant function expression

For \(x\) approaching 0, the relevant part of the function is \(-\frac{3}{2} x + \frac{1}{4}\). Since function behavior at a specific point \(x=0\) does not determine the limit, evaluate the expression assuming values are close to but not necessarily equal to 0.

Step 3: Evaluate the limit of the expression

Now, compute \(\lim _{x \rightarrow 0} (-\frac{3}{2} x + \frac{1}{4})\). Substituting \(x = 0\), we get:\[\begin{align*}\lim_{x \rightarrow 0} \left(-\frac{3}{2} x + \frac{1}{4}\right) & = -\frac{3}{2}(0) + \frac{1}{4} \& = 0 + \frac{1}{4} \& = \frac{1}{4}\end{align*}\]This gives us the limit value.

Key Concepts

Piecewise Functions

A piecewise function is a type of function defined by different expressions depending on the input value within certain intervals. It essentially "pieces" together multiple functions, forming a whole. Think of it like designing a patchwork quilt, where each patch serves a purpose but with individual conditions.Here's how it works:

• Each "piece" of the function is valid over a specific range of inputs.
• The function switches between different expressions based on these input values.
• This allows modeling of complex scenarios where a single formula might not suffice.

For instance, consider our function in the problem:- It has two pieces: 1. \(-\frac{3}{2}x + \frac{1}{4} \) valid for all x except when \(x = \frac{1}{2}\) 2. Value is 0 specifically when \(x = 0\)Unaffected by complexities like continuity at a single point, evaluating the pieces that overlap where they connect is crucial, especially when finding such function limits.

Limit Evaluation

Evaluating limits involves finding what value a function approaches as the input gets close to a certain point. This is a critical concept in calculus, allowing the analysis of behaviors of functions near specific values, often revealing insights not evident from direct substitution.To evaluate a limit, especially in piecewise functions, focus on the part relevant to the approaching value. For our example:

• The limit \( \lim_{x \to 0} f(x) \) is being sought.
• While \((x = 0)\) gives a definite value from the piecewise definition, it does not contribute to the limit.
• Investigate the behavior of \(-\frac{3}{2}x + \frac{1}{4}\) as \(x\) nears 0 from either direction (from the right or left).

Substituting directly in this component when approaching zero elucidates the function's continuity at this connecting point.

Function Substitution

Function substitution is a basic technique where you replace the variable with another value to determine behavior at different points. This is especially useful in solving limits.For the problem:

• First identify which piece of the piecewise function you're considering.
• When \(x\to 0\), the expression to evaluate is \(-\frac{3}{2}x + \frac{1}{4}\).
• Substitute \(x = 0\) into the equation: \(-\frac{3}{2} \times 0 + \frac{1}{4} = \frac{1}{4}\).

This reveals that as \(x\) gets closer to 0, the function indeed approaches a limit of \(\frac{1}{4}\).Substitution isn't just about plugging in numbers but aids in determining a pattern of limit behavior, helping to ascertain whether it tends towards a finite result or perhaps diverges.