Problem 26 - Precalculus: Mathematical for Calculus

Question

Graph the piecewise-defined function and use your graph to find the values of the limits, if they exist. $$f(x)=\left\\{\begin{array}{ll} 2 x+10 & \text { if } x \leq-2 \\ -x+4 & \text { if } x>-2 \end{array}\right.$$ (a) \(\lim _{x \rightarrow-2^{-}} f(x)\) (b) \(\lim _{x \rightarrow-2^{+}} f(x)\) (c) \(\lim _{x \rightarrow-2} f(x)\)

Step-by-Step Solution

Answer

The limits are: (a) 6, (b) 6, (c) 6.

Step 1: Interpret the Piecewise Function

The given function is piecewise defined, so we need to consider its definition depending on the value of \(x\). For \(x \leq -2\), the function is \(f(x) = 2x + 10\). For \(x > -2\), the function is \(f(x) = -x + 4\).

Step 2: Graphing the Function for x ≤ -2

For \(x \leq -2\), the function \(f(x) = 2x + 10\) is linear. Calculate few points: \((-2, 6)\), \((-3, 4)\). This segment continues with a closed circle at \(x = -2\).

Step 3: Graphing the Function for x > -2

For \(x > -2\), the function \(f(x) = -x + 4\) is also linear. Calculate few points: \((-1, 5)\), \((0, 4)\). This segment starts with an open circle at \(x = -2\).

Step 4: Analyze Limit at x approaches -2 from the Negative Side

For \(\lim_{x \to -2^{-}} f(x)\), substitute \(x = -2\) in \(2x + 10\). The value is \(2(-2) + 10 = 6\).

Step 5: Analyze Limit at x approaches -2 from the Positive Side

For \(\lim_{x \to -2^{+}} f(x)\), substitute \(x = -2\) in \(-x + 4\). The value is \(-(-2) + 4 = 6\).

Step 6: Evaluate the Two-Sided Limit at x = -2

If both one-sided limits are equal and exist (which they are, 6 in both cases), the limit of the function as \(x\) approaches \(-2\) exists and is 6.

Key Concepts

Limits of functions

When working with piecewise-defined functions, understanding limits is crucial to determine the behavior of the function at specific points. When you want to find the one-sided limits, you consider the function's behavior as it approaches a number from

• the left, written as \( \lim_{{x \to a^{-}}} f(x) \)
• the right, written as \( \lim_{{x \to a^{+}}} f(x) \)

In this particular exercise, you find both one-sided limits as \( x \rightarrow -2 \). For \( x \rightarrow -2^{-} \), you use the function defined for \( x \leq -2 \):

• Substitute -2 into \( 2x + 10 \) to find the limit from the left: \( 6 \).

Similarly, the limit from the right, \( x \rightarrow -2^{+} \), uses the second part of the function:

• Substitute -2 into \(-x + 4\), giving the limit from the right also as \( 6 \).

The two-sided limit exists if the left and right limits are equal. In this case, since both one-sided limits are 6, \( \lim_{x \to -2} f(x) = 6 \). This means that as \( x \) gets closer to \(-2\) from both directions, the function approaches the same value.

Graphing functions

Graphing piecewise-defined functions can seem tricky, but it's all about understanding each part separately. You have to graph the different pieces on their respective intervals:

• For the segment where \( x \leq -2 \), you sketch the graph of \( f(x) = 2x + 10 \). It's a simple linear function. Generate a few points, like (-2, 6) and (-3, 4), to plot.
• This part of the graph has a closed circle at \( x = -2 \), indicating that the value is included.
• Next, for \( x > -2 \), you switch to the second function piece, \( f(x) = -x + 4 \). Again, it is linear. Points like (-1, 5) and (0, 4) will help in plotting.
• The line will start with an open circle at \( x = -2 \), showing the value is not included here.

Once both parts are plotted, you will get a full visual representation of the function. Make sure to join the parts smoothly while respecting the circles; this formation clearly shows how the function behaves across the entire range of \( x \). Through this, the graph serves not just as a visual aid but also supports understanding how limits work with piecewise functions.

Continuity of functions

In mathematical terms, a function is continuous at a point if there is no interruption or jump when tracing the graph around that point. For a function to be continuous at a specific value \( a \), three conditions need to be met:

• The function is defined at \( a \), meaning \( f(a) \) exists.
• The limit of the function as \( x \) approaches \( a \) exists.
• \( \lim_{x \to a} f(x) = f(a) \), meaning the function's limit equals its value.

In our exercise, we analyze the point \( x = -2 \). The piecewise function is defined there, as seen by the closed circle from \( f(x) = 2x + 10 \).
Since the left-hand and right-hand limits both equal 6, they meet the second requirement. So, we conclude that:

• Though the two-sided limit \( \lim_{x \to -2} f(x) \) is 6, continuity also depends on \( f(-2) \), which is also 6. This matches the limit point, confirming the function is continuous at \( x = -2 \).

Understanding continuity in piecewise functions is often about checking where each piece begins or ends. This helps in recognizing any jumps or discontinuities in the graph.