Problem 41 - Essential Calculus

Question

In Exercises 41 and \(42,\) find the limit. (Hint: Let \(x=1 / t\) and find the limit as \(t \rightarrow 0^{+}\) ) $$ \lim _{x \rightarrow \infty} x \sin \frac{1}{x} $$

Step-by-Step Solution

Answer

The limit of \(x \sin \frac{1}{x}\) as \(x \rightarrow \infty\) is 1.

Step 1: Substitute \(x\) with \(1/t\)

Replace \(x\) in the function with \(1/t\) to get \((1/t) \sin t\). So the problem is now to find \( \lim _{t \rightarrow 0^{+}} (1/t) \sin t \)

Step 2: Recognize the limit as a known limit

If we look closely, we see that the function \((1/t) \sin t\), as \(t \rightarrow 0^{+}\), is in the form that can be compared to the standard limit \(\lim _{x \rightarrow 0} \frac{\sin x}{x} = 1\). We can rewrite our function as \(\sin t / t\), which is similar to \(\frac{\sin x}{x}\)

Step 3: Apply the standard limit

Since our function matches with the standard limit, we can apply this limit. Therefore, \( \lim _{t \rightarrow 0^{+}} \frac{\sin t}{t} = 1\)

Key Concepts

Calculus

Calculus is an essential branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. The concept of a limit is fundamental in calculus and serves as the foundation for defining derivatives and integrals. Limits describe the behavior of functions as they approach a certain point or infinity. Calculus enables us to study changes in functions and has applications in science, engineering, economics, statistics, and many other fields.

In the context of our exercise, calculus aids in understanding the behavior of the function as the variable tends toward infinity. By introducing a substitution technique, which is a common strategy in calculus, we simplify the limit evaluation. Using limits, we can deduce the behavior of complex expressions and determine their tendency as variables approach specific values or infinity.

Limits and Continuity

The limit of a function is a fundamental concept in calculus that deals with the value that a function approaches as the input approaches some value. Limits play a crucial role in defining continuity, derivatives, and integrals. In our exercise, the limit we are evaluating is as the variable approaches infinity, which often involves understanding the behavior of functions that grow without bound.

A function is said to be continuous at a point if the limit of the function as it approaches the point from both directions equals the function's value at that point. Continuity ensures that small changes in the input result in small changes in the output. The exercise provided demonstrates continuity by evaluating a limit where the input variable becomes infinitely large, a concept that can sometimes lead to counterintuitive results. Mathematically speaking, if a function is continuous at every point in an interval, then it is continuous on that interval.

Trigonometric Limits

Trigonometric limits are limits involving trigonometric functions, which often require special techniques or the application of known limits. The most famous trigonometric limit is \( \[ \lim _{x \rightarrow 0} \frac{\sin x}{x} = 1 \] \), which is frequently used to evaluate more complex limits involving sine functions.

In our exercise, by substituting \( x = \frac{1}{t} \) and changing the limit as \( t \rightarrow 0^{+} \) instead of \( x \rightarrow \infty \) and recognizing that we face a form of the standard trigonometric limit, we simplify the problem. Understanding how to manipulate a function and apply known trigonometric limits is key to solving many problems in calculus. The limit as \( t \rightarrow 0^{+} \) of \( \frac{\sin t}{t} \) is indeed 1, and this insight helps to determine the behavior of trigonometric functions as their arguments approach zero or other critical values.