Question

In Exercises \(21-26,\) find \(\lim _{x \rightarrow \infty} y\) and \(\lim _{x \rightarrow-\infty} y\). $$y=\frac{2 x+\sin x}{x}$$

Short answer

The limit of the function \(y=\frac{2 x+\sin x}{x}\) as \(x\) approaches infinity and negative infinity is 2

Step-by-step solution

Divide Each Term by x

Divide each term by x in the equation: \(y =\frac{2x}{x} + \frac{\sin x}{x} = 2 + \frac{\sin x}{x}\). This simplifies the equation and separates the terms making it easier to find the limits

Find the Limit as \(x\) Approaches Infinity

Now, we need to find the limit as \(x\) approaches infinity. For the first term, which is 2, the limit is simply 2 because it's a constant. For the second term, we know that \(-1 \leq \sin x \leq 1\) for all \(x\), so \(-\frac{1}{x} \leq \frac{\sin x}{x} \leq \frac{1}{x}\). As \(x\) approaches infinity, \(\frac{1}{x}\) and \(-\frac{1}{x}\) both approach zero. So, by the Squeeze theorem, \(\lim _{x \rightarrow \infty} \frac{\sin x}{x} = 0\). Hence, \(\lim _{x \rightarrow \infty} y = 2 + 0 = 2\)

Find the Limit as \(x\) Approaches Negative Infinity

Similarly, we can find the limit as \(x\) approaches negative infinity. The determination for the second term is the same as in Step 2. Hence, \(\lim _{x \rightarrow -\infty} y = 2 + 0 = 2\)

Key concepts

Calculus

In the broad field of mathematics, calculus stands as a monumental pillar, known for its role in formalizing concepts of change and motion. It is divided primarily into two branches: differential calculus and integral calculus. Differential calculus focuses on the rate of change, using derivatives to find how a function changes at any given point. Integral calculus, on the other hand, deals with accumulation of quantities, using integrals to find the total size or value, such as areas under curves.

Diving deeper into the subject, we find the concept of limits, which is foundational to both branches of calculus. Understanding limits allows us to examine the behavior of functions as they approach a certain value, which may not always be within the function's domain. For instance, limits at infinity are concerned with the behavior of a function as the input grows without bound, towards positive or negative infinity.

Squeeze Theorem

The Squeeze Theorem, also known as the Sandwich Theorem, is an essential part of calculus, particularly useful when evaluating limits of functions that are difficult to handle by direct substitution. It states that if a function 'f(x)' is always caught between two other functions 'g(x)' and 'h(x)' that have the same limit at a certain point, then 'f(x)' must also have that same limit at that point.

For example, if we know that for all 'x' in the interval, we have g(x) \<= f(x) \<= h(x), and also \text{lim}_{x \to c} g(x) = \text{lim}_{x \to c} h(x) = L, then it follows that \text{lim}_{x \to c} f(x) = L. This theorem is powerful in proving that certain trigonometric functions tend towards zero as x approaches infinity, as illustrated in the example where the squeeze theorem is applied to the function \(\frac{\sin x}{x}\).

Limit Properties

The calculus of limits is equipped with a toolkit of properties that help in calculating limits more conveniently. Among these are the limit laws that essentially allow mathematicians to break down complex expressions into simpler parts, each with their own limits. These laws include the sum law, product law, quotient law, and power law, to name a few.

For instance, when finding the limit of a sum, one can usually find the limits of the individual summands separately. In the given exercise, the function \(y = \frac{2x + \sin x}{x}\) is split into two terms — a linear term and a trigonometric term over 'x'. This separation is a direct application of these limit laws, simplifying the process and allowing us to evaluate each limit individually due to the additive property of limits.

Trigonometric Functions

Trigonometric functions are a class of functions that are crucial in the study of triangles, periodic phenomena, and much more. The primary trigonometric functions include sine (sin), cosine (cos), and tangent (tan), alongside their reciprocals cosecant (csc), secant (sec), and cotangent (cot).

In the context of limits and calculus, these functions often exhibit predictable behavior as their inputs become very large or very small. For sine and cosine, their values are always constrained between -1 and 1, regardless of the input value. This behavior is vital when evaluating limits at infinity for functions involving trigonometric terms, as shown in the provided exercise, where the limit of sin(x)/x is determined as x approaches infinity.