Problem 1 - College Algebra and Calculus: An Applied Approach

Question

Complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. $$ \lim _{x \rightarrow 2}(2 x+5) $$ $$ \begin{array}{|l|l|l|l|l|l|l|l|} \hline x & 1.9 & 1.99 & 1.999 & 2 & 2.001 & 2.01 & 2.1 \\ \hline f(x) & & & & ? & & & \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The limit as x approaches 2 of the function \(f(x) = 2x + 5\) is 9

Step 1: Understand the Function

The function given in this problem is \(f(x) = 2x + 5\). It's a straightforward linear function.

Step 2: Calculate Values of the Function

Plug in the values provided in the table for x into the function. After plugging in the values that are less than 2 (1.9, 1.99, 1.999), you should see that the value of the function approaches 9. Then plug in the values that are greater than 2 (2.001, 2.01, 2.1). You should also see that the function continues to approach 9.

Step 3: Estimating the Limit

After having calculated the value of functions at given points, it can be observed that as x gets arbitrarily close to 2, \(f(x)\) seems to be converging towards 9. Thus, the limit of the function as x approaches 2 can be estimated to be 9.

Step 4: Graph the Function

Plot the function \(f(x) = 2x + 5\). As x approaches 2, confirm visually that the value of the function is indeed approaching 9.

Key Concepts

Linear Function Limit

When estimating the limit of a linear function like the one given by the equation \( f(x) = 2x + 5 \) as \( x \) approaches a certain value, we're working to find out what \( f(x) \) is getting closer to as \( x \) gets nearer and nearer to that number.

In the case of a linear function, this process is relatively straightforward because linear functions have a constant rate of change—meaning they have no curves or bends. So, as \( x \) gets infinitely close to 2 in this problem, we'd expect \( f(x) \) to be getting infinitely close to a particular value, which is what we call the limit. Logic implies that, since a linear function is a straight line, the limit at any point along this line should simply be its corresponding function value. Thus, the estimated limit of this linear function as \( x \rightarrow 2 \) is \( f(2) \), which is indeed 9, reflecting the linearity and predictability of such functions.

To further solidify this concept, imagine sliding along the straight line graph of \( f(x) \) towards \( x = 2 \). No matter the direction from which you approach, the value of \( f(x) \) won't 'jump' or 'dip'; it will glide in a smooth transition toward that limit value.

Graphical Limit Confirmation

While calculating limits algebraically is useful, sometimes a graphical representation can be even more intuitive. Graphing the function \( f(x) = 2x + 5 \) and observing the behavior as \( x \) approaches 2 allows for visual confirmation of our calculated limit.

Visualizing the Approach

To confirm our estimated limit graphically, you can plot the values of \( f(x) \) on a graph for \( x \) around 2. You'll notice that as the plotted points draw near to \( x = 2 \), they will draw closer to the \( y \) value of 9. This gives a visual reinforcement of the concept of the limit. The graphical approach doesn't just serve as confirmation; it makes the concept more tangible.

Understanding Continuity

In addition, a graph helps in understanding another important concept in calculus—continuity. Since the graph of \( f(x) \) doesn't have any breaks, holes, or jumps, we can say our function is continuous at \( x = 2 \) and thus the limit visually meets the value of the actual function at that point.

Calculating Function Values

Part of mastering limits is being proficient at calculating function values, especially around the point of interest. Let's think of this process as putting pieces of a puzzle together to see a clearer picture of what's going on with the function as \( x \) gets close to a specific value.

By substituting various values of \( x \) into our function \( f(x) = 2x + 5 \) that are close to 2, we obtain function values that give us a snapshot of \( f(x) \) in that neighborhood. Here's the kicker: as the chosen \( x \) values get closer to 2, the corresponding \( f(x) \) values will get closer to \( f(2) \)—providing us with a clear indication of the limit from a numerical perspective.

To be thorough in understanding and accuracy, we make these calculations for points on both sides of 2—less than and greater than—to make sure the function is approaching the same value from both directions, which indeed it does: \( f(x) \) approaches 9.