Problem 25 - Calculus: Graphical, Numerical, Algebraic

Question

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

Step-by-Step Solution

Answer

The limit as \(x\) approaches both positive and negative infinity for the function \(y=\frac{\sin x}{2 x^{2}+x}\) is 0.

Step 1: Identify the structure

Firstly, examine the function \(y=\frac{\sin x}{2 x^{2}+x}\). Notice that as \(x\) approaches either positive or negative infinity, the degree of \(x\) in the denominator is higher than that in the numerator. The numerator value (\(\sin x\)) oscillates between -1 and 1.

Step 2: Apply the property of limits

For a rational function where the degree of the denominator is larger than the degree of the numerator, the limit as \(x\) tends to either positive infinity or negative infinity is 0. This is because as \(x\) grows larger, the denominator will grow much faster than the numerator, leading the whole fraction to approach 0.

Step 3: Solve for limit

Applying the property, we get \(\lim _{x \rightarrow \infty} y=\lim _{x \rightarrow-\infty} y=0\). As \(x\) approaches positive or negative infinity, the value of \(y\) will approach 0.

Key Concepts

Rational Functions

A rational function is defined as the ratio of two polynomials. Specifically, it has the form
\( f(x) = \frac{P(x)}{Q(x)} \)
where \( P(x) \) and \( Q(x) \) are polynomials. The key characteristics of a rational function include its domain, which is all real numbers except where the denominator, \( Q(x) \) is zero, and its behavior at these points, which often involve vertical asymptotes.

Rational functions can exhibit different behaviors as \( x \) approaches infinity, which is essential when finding limits. For instance, if the highest power of \( x \) in the denominator is greater than the highest power in the numerator, as \( x \) grows, the denominator increases much faster than the numerator, causing the value of the function to approach zero. This is exactly what occurs in the exercise where \(y = \frac{\sin x}{2x^2 + x} \) has a higher degree in the denominator.

Properties of Limits

Understanding the properties of limits is crucial when analyzing the behavior of functions as \( x \) approaches a certain value. One of these properties is the limit of a quotient. In the case of rational functions, if the numerator remains bounded (its value doesn't increase indefinitely) while the denominator grows without bound (approaches infinity), the quotient approaches zero.

Another important property is the squeeze theorem, which sometimes can be used to find the limits of functions with oscillating numerators, such as \( \sin x \) or \( \cos x \) when \( x \) increases without bound. In our exercise, as \( x \) approaches infinity or negative infinity, the numerator \( \sin x \) oscillates, but since it's bounded between -1 and 1, and the denominator grows to infinity, the value of \( y \) will be squeezed to zero. This is how we apply this property to deduce that both \( \lim _{x \rightarrow \infty} y \) and \( \lim _{x \rightarrow -\infty} y \) are zero.

Asymptotic Behavior

The concept of asymptotic behavior is used to describe how a function behaves as its argument gets very large or very small - in other words, how a function behaves near the 'ends' of the x-axis or near significant points on its graph. In layman terms, the function gets closer and closer to a certain line, called an asymptote, without actually touching it.

A common occurrence with rational functions is the presence of horizontal asymptotes, which show the behavior of the function as \( x \) tends to infinity or negative infinity. If the degree of the polynomial in the numerator is less than the degree of the polynomial in the denominator, the x-axis (\( y=0 \)) is typically the horizontal asymptote. For the function in the exercise \( y = \frac{\sin x}{2x^2 + x} \) the horizontal asymptote is \( y=0 \) because, as \( x \) approaches infinity or negative infinity, the value of \( y \) becomes arbitrarily small - approaching zero. This illustrates the function's asymptotic behavior.