Problem 55 - Calculus A New Horizon

Question

(a) Determine whether the following limits exist, and if so, find them: \(\lim _{x \rightarrow+\infty} e^{x} \cos x, \quad \lim _{x \rightarrow-\infty} e^{x} \cos x\) (b) Sketch the graphs of \(y=e^{x}, y=e^{-x},\) and \(y=e^{x} \cos x\) in the same coordinate system, and label any points of intersection. (c) Use a graphing utility to generate some members of the family \(y=e^{a x} \cos b x(a>0 \text { and } b>0),\) and discuss the effect of varying \(a\) and \(b\) on the shape of the curve.

Step-by-Step Solution

Answer

(a) The first limit does not exist; the second is 0. (b) Graphs show exponential growth/decay with oscillations. (c) Varying \(a\) and \(b\) affects amplitude and frequency.

Step 1: Evaluate Limit as x approaches +∞

The expression given is \( e^{x} \cos x \). When \( x \to +\infty \), \( e^{x} \to +\infty \). However, \( \cos x \) oscillates between -1 and 1. Thus, the product \( e^{x} \cos x \) becomes unbounded and keeps oscillating between \(-\infty\) and \(+\infty\). Therefore, the limit does not exist.

Step 2: Evaluate Limit as x approaches -∞

For the expression \( e^{x} \cos x \), when \( x \to -\infty \), \( e^{x} \to 0 \) because \( e^{x} = \frac{1}{e^{-x}} \) and \( e^{-x} \to +\infty \). \( \cos x \) remains bounded between -1 and 1. Thus, \( e^{x} \cos x \to 0 \times \cos x = 0 \) as \( x \to -\infty \). Therefore, the limit is 0.

Step 3: Sketch Graphs

The function \( y = e^{x} \) is an exponential growth function, which starts near zero for \( x \to -\infty \) and rises sharply as \( x \to +\infty \). The function \( y = e^{-x} \) is a decaying function, beginning high for \( x \to -\infty \) and dropping towards zero as \( x \to +\infty \). The function \( y = e^{x} \cos x \) combines exponential growth with oscillations due to \( \cos x \), resulting in waves of increasing amplitude as \( x \to +\infty \), and diminishing waves as \( x \to -\infty \), approaching 0.

Step 4: Use Graphing Utility to Explore Family of Functions

Using a graphing utility, you can plot \( y = e^{ax} \cos bx \) for different values of \(a\) and \(b\). Increasing \(a\) will amplify the rate of exponential growth or decay, thus affecting the amplitude of oscillations. Increasing \(b\) will increase the frequency of oscillations, leading to more peaks and troughs per unit \(x\). Lower values flatten the curve and reduce wave frequency.

Key Concepts

Exponential Functions

Exponential functions, like the ones explored in this problem, are mathematical expressions of the form \( y = e^{x} \) where \( e \) is the mathematical constant approximately equal to 2.718, and \( x \) is the variable exponent. These functions are known for their distinctive growth patterns.
For positive values of \( x \), as \( x \) goes to infinity, \( e^{x} \) grows extremely rapidly. This is because each increment in \( x \) leads to the entire value multiplying based on the number \( e \), which is why exponential functions are often depicted as a curve that initially appears flat and then shoots upwards steeply.
Conversely, for negative values of \( x \), the expression \( e^{x} \) approaches zero. It's important to note how this behavior significantly impacts the outcome when combined with different components, such as the oscillatory function \( \cos x \) in this exercise.
Exponential functions often appear in real-world scenarios where growth processes exhibit a continuous, accelerating change, like population growth, radioactive decay, or interest calculations.

Oscillatory Functions

Oscillatory functions, particularly trigonometric ones like \( \cos x \), are key players in mathematics, characterized by their repetitive and cyclic nature. The cosine function oscillates between a maximum value of 1 and a minimum value of -1. This implies that no matter how much \( x \) increases or decreases, \( \cos x \) will continue to "wiggle" within this limited range.

• The fundamental period of \( \cos x \) is \( 2\pi \), meaning it completes a full cycle from peak to peak over this interval.
• In the problem at hand, even though \( e^{x} \) might tend to infinity, the oscillatory behavior of \( \cos x \) can cause the product to behave unpredictably when combined with a rapidly growing exponential function.
• These functions are crucial in modeling wave behaviour, describing things like sound and light waves, and can also be seen in seasonal patterns or cyclical economic trends.

Graphical Analysis

Graphical analysis of equations like \( y = e^{x} \), \( y = e^{-x} \), and \( y = e^{x} \cos x \) allows us to visualize how mathematical concepts unfold in a visual format. Graphing these functions provides crucial insights into the nature of their behavior.
By plotting \( y = e^{x} \), one can observe the steep exponential increase, making it clear how data, growth, or values are sharply rising as \( x \) becomes large. Alternatively, \( y = e^{-x} \) showcases a rapid decline toward zero, highlighting a common decay process.
The function \( y = e^{x} \cos x \) offers a fascinating study in combined behaviors. The exponential component \( e^{x} \) attempts to pull the function values toward infinity, while the oscillatory component \( \cos x \) continuously changes from -1 to 1, resulting in waves of growing amplitude for \( x \to +\infty \) and diminishing as \( x \to -\infty \).
Exploring the family of functions \( y = e^{a x} \cos b x \) through graphing allows us to further manipulate and observe how variations in the parameters \( a \) and \( b \) redefine the shape and nature of the curve, crucial for predicting or simulating differing real-world phenomena.