Question

One-Sided Limit In Exercises \(49-52\) , use a graphing utility to graph the function and determine the one-sided limit. $$ \begin{array}{l}{f(x)=\frac{1}{x^{2}-25}} \\ {\lim _{x \rightarrow 5^{-}} f(x)}\end{array} $$

Short answer

The one-sided limit, or \(\lim _{x \rightarrow 5^{-}} f(x)\), will be determined from the observation of the graph of the function \(f(x) = \frac{1}{x^{2} - 25}\) as \(x\) approaches \(5\) from the left-hand side. The exact value or conclusion (like 'the limit is positive/negative infinity' or 'the limit does not exist') depends on the particular graph.

Step-by-step solution

Graph the Function

First, plot the function \(f(x) = \frac{1}{x^{2} - 25}\) using any graphing utility tool. Carefully observe the behaviour of the function when \(x\) is close to \(5\) on the left-hand side of the number line.

Locate x=5 on the Graph

Locate the point on the graph where \(x = 5\). Mark this point. Now move toward this point from the left side (i.e., as \(x\) approaches \(5\) from values less than \(5\)). Observe how the function behaves.

Determine the One-Sided Limit

The one-side limit, \(\lim _{x \rightarrow 5^{-}} f(x)\) is the value that \(f(x)\) approaches as \(x\) approaches \(5\) from the left-hand side. If the graph approaches a certain value, that's your limit. If the function value increases or decreases without bound, the limit is positive or negative infinity. If there is no unique value the function is heading toward, the limit does not exist.

Key concepts

Graphing Utility and Its Importance

A graphing utility is a digital tool that helps visualize mathematical functions and their behaviors. These can include software programs or graphing calculators. Using a graphing utility, you can easily plot functions like \( f(x) = \frac{1}{x^{2} - 25} \).
Seeing a function's graph helps in understanding how it behaves over different intervals of the domain. For instance, it shows where the function increases, decreases, or if it has any asymptotes. Graphing utilities are valuable because they provide:

• Visual representations of abstract mathematical concepts.
• Quick and accurate ways to analyze and interpret functions.
• A way to explore functions without solving complex algebraic equations manually.

To use a graphing utility, you input the function equation, and the tool generates the graph almost instantly. The ability to adjust settings such as the window range allows you to focus on different sections of the graph, which is essential for analyzing specific behavior, like one-sided limits.
This visualization supports deeper understanding and can help identify points of interest like limits, intercepts, and asymptotes.

Understanding Left-Hand Limit

A left-hand limit refers to the behavior of a function as the variable approaches a certain point from the left on the number line. Using the graph for the function \( f(x) = \frac{1}{x^{2} - 25} \), we are interested in the limit as \( x \) approaches \( 5 \) from the left, denoted as \( \lim_{x \to 5^-} f(x) \).
To find this limit, you focus on the values of \( x \) that are slightly less than 5 but getting closer and closer to it from the left side. By closely observing the graph, you assess:

• If the function f(x) approaches a particular value, the left-hand limit is that value.
• If the function values rise or fall without bound as \( x \) nears 5 from the left, then the left-hand limit might be positive or negative infinity.
• If no specific value is approached, the limit does not exist.

This approach provides a visual and intuitive understanding of limits, making concepts that are often abstract much more tangible.

Analyzing Limit Behavior

Limit behavior describes how a function behaves near a specific point or as the input increases indefinitely. In the exercise provided, we determine the behavior of \( f(x) = \frac{1}{x^{2} - 25} \) as \( x \) tends toward 5 from the left.
By using a graph, you plot the function and observe the trend as \( x \to 5^- \):

• If the graph suggests the function approaches a finite number, that is the limit.
• If the function value climbs or drops excessively, the limit may be infinite.
• Sometimes, a function does not settle into a single value but rather oscillates or jumps unpredictably, indicating no limit.

Understanding these behaviors requires examining the graph closely around the chosen point. By doing so, you can anticipate how the function behaves not just at this point but also potentially throughout its domain. This insight is useful in broader applications like calculus and problem-solving scenarios where predicting a function's behavior is critical.

Function Analysis Insights

Function analysis is the comprehensive study of the properties and behavior of functions. In our specific analysis of the function \( f(x) = \frac{1}{x^{2} - 25} \), we're focused on its one-sided limit. However, function analysis typically includes examining:

• Domain and range: Where the function is defined and its possible output values.
• Continuity: Whether the function has any breaks, holes, or asymptotes.
• Asymptotic behavior: How the function behaves as it heads towards infinity or a vertical asymptote.
• Critical points: Points where the function's derivative is zero or undefined, indicating potential maxima, minima, or inflection points.

This thorough investigation helps in understanding how changes in the input affect the output, and it provides tools for predicting future behavior. For the limit \( \lim_{x \to 5^-} f(x) \), it is crucial because it reveals how the function responds as it nears the vertical asymptote at \( x = 5 \).
Understanding these fundamentals aids students and mathematicians alike in solving complex problems and conducting advanced analyses, making function analysis an invaluable skill.