Problem 19 - A Graphical Approach to Precalculus with Limits

Question

Use the table of values to predict \(\lim _{x \rightarrow 1} f(x)\) $$\begin{array}{|c|c|c|c|c|c|c|} \hline x & 0.9 & 0.99 & 0.999 & 1.001 & 1.01 & 1.1 \\ \hline f(x) & 1.9 & 1.99 & 1.999 & 2.001 & 2.01 & 2.1 \end{array}$$

Step-by-Step Solution

Answer

\(\lim_{x \to 1} f(x) = 2\)

Step 1: Analyze the Table Values

Check the values of \(f(x)\) as \(x\) approaches 1 from both sides. From the given table, observe that as \(x\) approaches 1 from the left (0.9, 0.99, 0.999), \(f(x)\) values (1.9, 1.99, 1.999) get closer to 2. Similarly, as \(x\) approaches 1 from the right (1.001, 1.01, 1.1), \(f(x)\) values (2.001, 2.01, 2.1) also converge towards 2.

Step 2: Identify the Limit

From Step 1, notice that \(f(x)\) is consistently moving closer to the number 2 from both sides of \(x = 1\). This behavior indicates that the function values are stabilizing around 2 as \(x\) approaches 1.

Step 3: Conclude the Limit

Since both the left-hand limit (as \(x\) approaches 1 from the left) and the right-hand limit (as \(x\) approaches 1 from the right) approach the same value, the two-sided limit exists and is equal to this value.

Key Concepts

Function Behavior

When studying limits, understanding function behavior is crucial. Knowing how a function behaves gives insight into how it approaches a particular point. Function behavior tells us how the output of a function reacts to changes in input values.
For example, as shown in the provided table, as the input values of \( x \) get closer to 1, the output \( f(x) \) gets closer to 2 from both sides.This pattern indicates that the function is smoothly nearing a stable value. Evaluating this behavior helps in predicting limits, even in cases where they might not be obvious at first glance.

Two-sided limits

Two-sided limits refer to the value that a function approaches as the input gets closer to a particular number from both directions.When calculating two-sided limits, we are interested in seeing if the function approaches the same value from the left and the right.
In the exercise example, checking two-sided limits involves observing the behavior of \( f(x) \) as \( x \) approaches 1 from both sides.The table shows that \( f(x) \) approaches 2 from both the left and right sides.
This observation confirms that a two-sided limit exists at this point, indicating the function's behavior is consistent.

Approaching Values

Approaching values are the key to understanding limits. These are the predicted values a function nears as the input variable approaches a specific point.
From the table, notice that as \( x \) gets closer to 1, the \( f(x) \) values also get closer to 2, from both directions.This suggests that the measured values are approaching a common value, which is the core essence of limits. Such observations are indispensable in predicting the behavior of functions around critical points.

Left-hand Limit

The left-hand limit is the value that a function approaches as the input variable approaches a certain point from the left (or smaller) side.It helps us understand how a function behaves specifically from one side.
In our example, as \( x \) approaches 1 from the left-hand side with values like 0.9, 0.99, and 0.999, \( f(x) \) approaches 2.This left-sided observation gives the partial picture of the overall limit and is crucial in confirming the two-sided limit.

Right-hand Limit

The right-hand limit considers values as the input variable approaches a particular point from the right (or larger) side.Understanding this helps in assessing how a function approaches a value from the higher side of a specific input.
From the table, we see as \( x \) approaches 1 from the right with values such as 1.001, 1.01, and 1.1, \( f(x) \) consistently moves towards 2.This consistency from the right side supports the determination of a unified two-sided limit, affirming that both one-sided limits lead to the same conclusion.