Question

Numerical, Graphical, and Analytic Analysis In Exercises \(49-52,\) use a graphing utility to complete the table and estimate the limit as \(x\) approaches infinity. Then use a graphing utility to graph the function and estimate the limit. Finally, find the limit analytically and compare your results with the estimates. $$ \begin{array}{|l|l|l|l|l|l|l|l|} \hline x & 10^{0} & 10^{1} & 10^{2} & 10^{3} & 10^{4} & 10^{5} & 10^{6} \\ \hline f(x) & & & & & & & \\ \hline \end{array} $$ $$ f(x)=x \sin \frac{1}{2 x} $$

Short answer

The limit of \( f(x) = x\sin(1/(2x)) \) as \( x \) approaches infinity is undefined.

Step-by-step solution

Table Representation

Use a graphing utility or scientific calculator to generate the numerical values of the function \(f(x) = x\sin(1/(2x))\) for each value of \(x\). This should be done for \(x = 10^0, 10^1, 10^2, 10^3, 10^4, 10^5, 10^6\).

Graphical Representation

Use a graphing utility to graph the function \(f(x) = x\sin(1/(2x))\). With the graph, one can estimate the limit as \(x\) approaches infinity. The limit is reflected by the value the curve approaches as \(x\) becomes extremely large.

Analytic limit calculation

Analytically, the limit can be found using the sandwich theorem. Observe that \( -1 \leq \sin(1/(2x)) \leq 1 \), thus, \(-x \leq x\sin(1/(2x))\leq x\). As \(x\) approaches infinity, \( -x \) goes to negative infinity and \( x \) goes to positive infinity. Because \(\sin(1/(2x))\) is bound by -1 and 1, we see that the limit of the function as \(x\) approaches infinity is not established, thus it is undefined.

Key concepts

Graphing Utility

When studying calculus, one of the valuable tools at your disposal is a graphing utility. This software allows you to visualize mathematical functions and their behavior. In the context of estimating limits, such as in the exercise given, students are encouraged to input the function into the graphing utility and observe the graph's behavior as the variable approaches a certain value, in this case, infinity.

A graph can provide an immediate intuitive grasp of a function's limit. As the values on the x-axis increase towards infinity, you can see whether the function's values are approaching a finite number, become unbounded, or oscillate without settling down. It's important to note that while graphing utilities provide a visual estimate, they are not always conclusive in determining limits, especially when it comes to infinity. This is why analytical methods are also necessary. Nonetheless, graphing is a strong starting point and can guide you to the correct analytical methods needed to find a limit.

Analytical Limit Calculation

The analytical calculation of limits serves as the mathematical bedrock for understanding function behavior. This method employs algebraic techniques to find the exact value that a function approaches as the variable approaches a particular point or infinity. Unlike estimates from a graphing utility, analytical methods can provide definitive answers.

In the case of the function f(x) = x sin(1/(2x)), analytical methods could involve evaluating the function's behavior as x grows large. By applying techniques from calculus, like the sandwich theorem or L'Hôpital's Rule, you obtain precise conclusions about the behavior of the function. It's a critical skill set for any student tackling calculus problems since it ensures accuracy and a deep comprehension of function limits as they approach any value, including infinity.

Limit as x Approaches Infinity

When we talk about the limit of a function as x approaches infinity, we are interested in understanding what value, if any, the function stabilizes at as x gets indefinitely large. This concept is crucial in calculus because it can tell us about the long-term behavior of functions which is important in many areas of mathematics and applied sciences.

However, not all functions have limits as x approaches infinity. For some functions, the values might keep increasing or decreasing without bound or oscillate in an unpredictable manner. The function in the exercise, f(x) = x sin(1/(2x)), as x increases, the sinusoidal component varies between -1 and 1. Multiplying by x, the function does not settle on a certain value but rather increases and decreases without bound, leading to the conclusion that the limit is undefined as x approaches infinity.

Sandwich Theorem

The Sandwich Theorem, also known as the Squeeze Theorem, is a critical concept in calculus for determining the limit of a function. This theorem tells us that if a function can be 'squeezed' between two other functions that have the same limit at a certain point, then the middle function must also approach that same limit at that point.

In the step-by-step solution, the Sandwich Theorem is applied by recognizing that the sine function oscillates between -1 and 1, and by sandwiching it with two linear functions -x and x. As x approaches infinity, both -x and x approach infinity, but in opposite directions, negative and positive, respectively. Since the sine function is bounded between these two, it can neither approach a finite limit nor can it become unbounded in one direction as x approaches infinity. Therefore, we conclude that the limit of f(x) as x approaches infinity is undefined. The Sandwich Theorem is an elegant way to analytically determine limits that might otherwise be challenging to ascertain.