Problem 43 - A Graphical Approach to Precalculus with Limits

Question

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value. \(\lim _{x \rightarrow 3} f(x),\) where \(f(x)=\left\\{\begin{array}{ll}x+7 & \text { if } x \leq 3 \\ 5 x-5 & \text { if } x>3\end{array}\right.\)

Step-by-Step Solution

Answer

The limit exists and is 10.

Step 1: Analyze the Function

Understand that we have a piecewise function. For \(x \leq 3\), the function is \(f(x) = x + 7\). For \(x > 3\), the function is \(f(x) = 5x - 5\). We will explore how this function behaves as \(x\) approaches 3 from the left and from the right.

Step 2: Evaluate the Left-Hand Limit

Since \(x\) is approaching 3 from the left (\(x \leq 3\)), use the expression \(f(x) = x + 7\). Substitute \(x = 3\) and calculate:\[\lim_{{x \to 3^-}} f(x) = 3 + 7 = 10.\]

Step 3: Evaluate the Right-Hand Limit

As \(x\) approaches 3 from the right (\(x > 3\)), use the expression \(f(x) = 5x - 5\). Substitute \(x = 3\) in the limit:\[\lim_{{x \to 3^+}} f(x) = 5(3) - 5 = 15 - 5 = 10.\]

Step 4: Compare Left-Hand and Right-Hand Limits

Both the left-hand limit and right-hand limit as \(x\) approaches 3 are equal to 10. \(\lim_{{x \to 3^-}} f(x) = \lim_{{x \to 3^+}} f(x) = 10.\)

Step 5: Conclude Limit Existence and Value

Since both the left-hand limit and right-hand limit are equal, the limit exists. Therefore, \(\lim_{{x \to 3}} f(x) = 10.\)

Key Concepts

Piecewise Functions

Piecewise functions are mathematical expressions which are defined by different formulas across a range of values in their domain. Essentially, they allow you to describe a situation in which a function takes different forms based on an input value.

• Each "piece" of the function applies to a specific interval of the domain.
• They provide great flexibility in modeling situations where behavior changes at certain points.
• In the given function, the expression changes at the point where \(x = 3\).

For example, with the function from the exercise:

• For \(x \leq 3\): the function is defined as \(f(x) = x + 7\).
• For \(x > 3\): it changes to \(f(x) = 5x - 5\).

Graphs of piecewise functions often have different segments that do not necessarily connect smoothly unless specified otherwise.

Left-Hand Limit

The left-hand limit, symbolized as \(\lim_{{x \to c^-}} f(x)\), describes the behavior of a function as "x" approaches a specified point "c" from the left (or from values smaller than "c").

• In practical terms, you evaluate the function using values just below "c".
• This concept helps understand the function's behavior near specific boundary points.

For the exercise at hand, the left-hand limit was calculated using the expression applicable when \(x \leq 3\), namely, \(f(x) = x + 7\). The limit as \(x\) approaches 3 from the left is:\[\lim_{{x \to 3^-}} f(x) = 3 + 7 = 10.\]By evaluating this, we gain insight into the function's behavior approaching that critical boundary.

Right-Hand Limit

The right-hand limit, denoted \(\lim_{{x \to c^+}} f(x)\), evaluates the behavior of a function as "x" approaches a specific point "c" from the right (or from values greater than "c").

• To find this limit, consider the function values just above "c".
• It complements the left-hand limit for understanding the full behavior of a function around discontinuities or transition points.

In the problem provided, the right-hand limit was determined using the piece for \(x > 3\), which is \(f(x) = 5x - 5\). The limit as \(x\) approaches 3 from the right calculates to:\[\lim_{{x \to 3^+}} f(x) = 5(3) - 5 = 15 - 5 = 10.\]Both the left and right-hand limits give us a view on whether a function transitions smoothly at the boundary, confirming continuity at that point if they are equal.