Problem 15 - A Graphical Approach to Precalculus with Limits

Question

Determine each limit. Refer to the accompanying graph of \(y=f(x)\) when it is given. Do not use a calculator. $$\lim _{x \rightarrow-3^{-}} \frac{|x+3|}{x+3}$$

Step-by-Step Solution

Answer

The limit is -1.

Step 1: Analyze the Expression

The expression is \( \frac{|x+3|}{x+3} \). This involves an absolute value function in the numerator. We need to break it down to evaluate the limit specifically as \( x \) approaches \(-3\) from the left, denoted by \( x \rightarrow -3^{-} \).

Step 2: Consider x Approaching -3 from the Left

Since we approach \(-3\) from the left (values of \(x\) slightly less than \(-3\)), \( x+3 \) will be negative. Thus, \( |x+3| = -(x+3) \).

Step 3: Substitute the Expression

Substitute \( |x+3| = -(x+3) \) into the fraction to get \( \frac{-(x+3)}{x+3} \).

Step 4: Simplify the Fraction

The expression \( \frac{-(x+3)}{x+3} \) simplifies to \(-1\) because \( (x+3) \) in the numerator and denominator cancels out, leaving \( -1 \).

Step 5: Determine the Limit

As \( x \rightarrow -3^{-} \), the simplified expression \( -1 \) remains constant, so \( \lim_{x \rightarrow -3^{-}} \frac{|x+3|}{x+3} = -1 \).

Key Concepts

Graphical Analysis

Understanding limits using graphical analysis helps visualize how a function behaves as it approaches a specific value. In this problem, we focus on analyzing the function behavior as \( x \) approaches -3 from the left. Graphically speaking, this means observing values of \( x \) that are just slightly less than -3.
To graphically assess this, imagine a graph with the x-axis and a point, say -3, nearing on the line. As \( x \) nears -3 from the left, it involves analyzing the leftward side trend of the graph for \( y = \frac{|x+3|}{x+3} \).
Consequently, you would see that the graph levels off at a constant value indicating the behavior of the function at that point. This is crucial when determining graphical trends or confirming algebraic results such as the limit being \(-1\).

Absolute Value Functions

Absolute value functions are designed to convert any input into its non-negative equivalent. This distinct characteristic becomes significant when solving limits involving absolute values, like in our problem.
The function in the numerator, \( |x+3| \), is crucial. Here, the absolute value affects the expression depending on whether \( x+3 \) is positive or negative. If \( x+3 \) is negative, \( |x+3| \) becomes \(-(x+3)\). This characteristic helps when analyzing movement towards a number, such as -3.
In the given exercise, since approaching from the left results in negative \( x+3 \), the absolute value simplifies and affects the final outcome. Understanding how absolute value functions influence behavior helps you reason through more complicated expressions.

One-sided Limits

One-sided limits focus on behavior from a specific direction, either from the left or the right. The notation \( x \rightarrow -3^{-} \) tells us we're dealing with how \( x \) approaches -3 from the left.
This means considering values smaller than \(-3\) but close to it, which changes how the expression \( \frac{|x+3|}{x+3} \) is approached. In this scenario, since the determinant part \( x+3 \) is negative, you see it affects the numerator first, resulting in a negative equivalent.
It simplifies to a clear, constant value, \(-1\), as shown in the problem. The concept of one-sided limits thus directs us to only consider specific directional behavior, ignoring the other side, making it easier to solve for particular circumstances.