Problem 45 - Contemporary Precalculus

Question

Find \(\lim _{h \rightarrow 0} \frac{f(0+h)-f(0)}{h},\) if it exists. $$f(x)=|x|$$

Step-by-Step Solution

Answer

#tag_title# Step 3: Compute the limit for both cases #tag_content# Let's compute the limit for each case separately. Case 1: x ≥ 0: $$\lim _{h \rightarrow 0} \frac{f(0+h)-f(0)}{h} = \lim _{h \rightarrow 0} \frac{h-0}{h} = \lim _{h \rightarrow 0} 1 = 1$$ Case 2: x < 0: $$\lim _{h \rightarrow 0} \frac{f(0+h)-f(0)}{h} = \lim _{h \rightarrow 0} \frac{-h-0}{h} = \lim _{h \rightarrow 0} (-1) = -1$$ Now, let's analyze the results. #tag_title# Step 4: Analyze the results and determine if the limit exists #tag_content# We found the limit in the two cases to be different: 1 for x ≥ 0 and -1 for x < 0. Since the limits are different for each case, the overall limit at x = 0 does not exist. So, the answer is that the limit of the difference quotient as h approaches 0 for the function \(f(x) = |x|\) does not exist.

Step 1: Write down the given function and the limit to be found

The given function is \(f(x) = |x|\). We need to find the limit $$\lim _{h \rightarrow 0} \frac{f(0+h)-f(0)}{h}$$.

Step 2: Separate the function into two cases

Since the absolute value function is piecewise defined, we need to separate it into two cases: $$f(x) = \begin{cases} x &\text{for} \ x \geq 0, \\ -x &\text{for} \ x < 0. \end{cases}$$ Step 3: Compute the limit for both cases

Key Concepts

Absolute Value

The absolute value of a real number refers to its distance from zero on the number line, regardless of direction. Mathematically, it is defined as:
\[\begin{equation} \ |x| =\begin{cases} x & \text{if } x \geq 0,\ -x & \text{if } x < 0.\ \end{cases}\end{equation}\]
This piecewise representation of absolute value is crucial to understanding how it behaves under operations like limits. When evaluating limits involving absolute values, we must consider both the non-negative and negative cases, as they can lead to different results. Failure to do so often results in an incomplete understanding of the function's behavior, especially around the origin where the piecewise definitions switch.

Piecewise Functions

Piecewise functions are defined by different expressions for different intervals of their domain. These functions can have a variety of appearances in their graph, and understanding each piece's behavior is key to evaluating limits and continuity. For instance, the absolute value function is inherently piecewise since it behaves differently for positive and negative inputs.

When finding limits for piecewise functions, separately investigate the behavior as the function approaches the point of interest from different directions. These so-called 'one-sided limits' are instrumental in determining the overall limit and are a common point of confusion if not handled with care. It's worth noting that not every discontinuity in a piecewise function is a 'jump'; it could also be a 'hole', or point of removable discontinuity, where the function is undefined despite having a clear limit.

Limits and Continuity

In calculus, limits are used to describe the behavior of a function as it approaches a particular point, which is essential in defining continuity. Continuity at a point means that the limit of the function as it approaches the point is equal to the function's value at that point.

For a function defined by absolute value, exploring limits and continuity often requires assessing the behavior from both sides of a point, and applying the definition of limits cautiously. In the example given, the function's adherence to continuity is tested at the point where the absolute value function 'switches' from one expression to another.
Such careful consideration ensures a comprehensive understanding of a function across its entire domain - leading to more accurate calculations and a more profound appreciation for the intricate behavior of piecewise functions like the absolute value.