Question

Evaluate \(\lim _{x \rightarrow 0} \frac{|3 x-4|}{x-2}\)

Short answer

Answer: The limit is -2.

Step-by-step solution

Find the interval for this case

First, we solve the inequality to find the interval of x values for which the first part of the piecewise function holds: $3x - 4 \geq 0 \\ 3x \geq 4 \\ x \geq \displaystyle\frac{4}{3}$

Evaluate the limit for this case

Now we analyze the limit of the first part of the piecewise function as \(x\) approaches 0: \(\lim_{x \rightarrow 0} \frac{3x - 4}{x - 2}\) However, since \(x \geq \displaystyle\frac{4}{3}\), this case does not apply when \(x\) approaches 0. Case 2 (When \(3x - 4 < 0\))

Find the interval for this case

Next, we solve the inequality for the interval of x values for which the second part of the piecewise function holds: $3x - 4 < 0 \\ 3x < 4 \\ x < \displaystyle\frac{4}{3}$

Evaluate the limit for this case

In this case, as \(x\) approaches 0, the second part of the piecewise function applies: \(\lim_{x \rightarrow 0} \frac{-(3x - 4)}{x - 2}\) To find this limit, we can apply the limit rules and simplify the function: \(\lim_{x \rightarrow 0} \frac{-3x + 4}{x - 2} = \frac{-3(0) + 4}{0 - 2} = \frac{4}{-2} = -2\)

Check if the limit exists

Now that we have evaluated the limit for the second case, we need to check if the limit exists. Because only the second case applies when \(x\) approaches 0, we can conclude that the limit does exist and: \(\lim_{x \rightarrow 0} \frac{| 3x - 4 |}{x - 2} = -2\)

Key concepts

Piecewise Functions

Piecewise functions are a unique type of function defined differently over multiple intervals. They let us "piece together" different segments that behave differently based on the input value, usually involving inequalities.
One segment might behave as a linear function when the variable exceeds a certain number, and another might behave differently below that number.
In the context of limits, analyzing piecewise functions involves checking each "piece" to see which applies as the input approaches a specific value. This often requires solving inequalities to determine valid intervals. For example, in the exercise above:

• We first check the inequality \(3x - 4 \geq 0\) to define one segment.
• Then, we solve \(3x - 4 < 0\) for another segment.

Once we identify the correct part of the function to evaluate as the variable approaches a certain value, we can proceed with determining the limit. It is crucial to follow these steps carefully as they ensure we use the correct function for the limit evaluation.

Limit Rules

Limit rules are fundamental in calculus, allowing us to evaluate limits systematically. They include various properties and methods that simplify the process of finding a limit.
When applying limit rules, such as in the exercise solution, we make the function simpler and easier to evaluate when the input approaches a particular value. For instance:

• Simplify the expression: transform complex fractions or expressions to simpler forms.
• Use direct substitution: once simplified, direct substitution helps find the limit value easily.

In the given example, we face an absolute value expression. It first requires us to determine the correct piece through inequalities, after which we apply the negative of the expression (due to it being valid for \(3x - 4 < 0\)).
Then, we simplify and use substitute where appropriate to find that \(\lim_{x \to 0} \frac{|3x - 4|}{x - 2} = -2\). Carefully applying these steps ensures the calculation is accurate and verifies the limit's existence for a particular interval.

Inequalities in Calculus

Inequalities play a key role in calculus, especially while working with piecewise functions and limits. They help determine when a specific function or expression becomes suitable for evaluating a limit as a variable approaches a point.
In many cases, such as evaluating limits, understanding inequalities involves:

• Identifying critical points by solving inequalities to define distinct intervals for a function.
• Determining which piece of a piecewise function to use based on these intervals.

For the example provided:

• The inequality \(3x - 4 < 0\) informs us that the second piece applies as \(x\) approaches 0.
• This ensures that we evaluate the right part, resulting in a correct limit calculation.

Effectively handling inequalities ensures that piecewise functions are correctly evaluated, leading to precise limit determination. Consequently, wisdom in solving inequalities is crucial for calculus success, especially when tackling exercises involving discontinuous functions or absolute values.