Problem 49 - Contemporary Precalculus

Question

Use numerical or graphical means to find the limit, if it exists. If the limit of f as x approaches c does exist, answer this question: Is it equal to \(f(c) ?\) $$\lim _{x \rightarrow 1} e^{x} \sin \frac{\pi x}{2}$$

Step-by-Step Solution

Answer

Answer: The limit of the function \(e^{x} \sin \frac{\pi x}{2}\) as x approaches 1 is e, and it does exist since \(\lim_{x \rightarrow 1} e^{x} \sin \frac{\pi x}{2} = f(1)\).

Step 1: Identify the limit notation

The exercise asks for the limit of the function \(e^{x} \sin \frac{\pi x}{2}\) as x approaches 1: $$\lim _{x \rightarrow 1} e^{x} \sin \frac{\pi x}{2}$$

Step 2: Evaluate the sin function at x=1

First, let's find the value of the sin function at x=1: $$\sin \frac{\pi (1)}{2} = \sin \frac{\pi}{2}$$ Recall that the sin function has a maximum value of 1 at \(\frac{\pi}{2}\). So, $$\sin \frac{\pi}{2} = 1$$

Step 3: Evaluate the exponential function at x=1

Next, let's evaluate the exponential function: $$e^{1} = e$$

Step 4: Find the limit by combining the evaluated functions

Combining the evaluated sin and exponential functions, we get: $$\lim _{x \rightarrow 1} e^{x} \sin \frac{\pi x}{2} = e \cdot 1 = e$$ So the limit as x approaches 1 is equal to e.

Step 5: Evaluate the function at x=1

Now, let's evaluate the given function at x=1 to see if it is the same as our limit: $$f(1) = e^{1} \sin \frac{\pi (1)}{2} = e \cdot 1 = e$$

Step 6: Compare the limit and the function value

Since the limit as x approaches 1 is e, and the function value at x=1 is also e, we can conclude that the limit exists and: $$\lim _{x \rightarrow 1} e^{x} \sin \frac{\pi x}{2} = f(1)$$

Step 7: Verify the limit graphically

Use a graphing calculator or software to graph the function \(e^x \sin \frac{\pi x}{2}\). Observe the graph as it approaches x=1. You should see that the graph does indeed approach the value e, confirming our analytical analysis.

Key Concepts

Exponential Functions

Exponential functions are a fundamental part of calculus and appear frequently in limit problems. The function shown here, \( e^x \), is a prime example. Here, \( e \) is the base of the natural logarithm, approximately equal to 2.718. In calculus, exponential functions are noteworthy because they grow rapidly; in fact, they grow faster than any polynomial as \( x \) increases. This growth is due to the nature of the exponential base, \( e \), which is unique for its property that the rate of growth of \( e^x \) is proportional to its current value.

When evaluating limits involving exponential functions, like \( \lim _{x \rightarrow 1} e^x \), inserting \( x = 1 \) directly into the equation can often yield the desired value, \( e \). For this exercise, the exponential function contributed directly and simply by evaluating as \( e^1 = e \), which integrates seamlessly into the overall limit evaluation.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are essential in calculus for understanding oscillatory behavior. In this exercise, we specifically dealt with the sine function, which oscillates between -1 and 1. The argument, \( \frac{\pi x}{2} \), transforms this familiar function as \( x \) varies.

To find limits involving trig functions, it is often useful to evaluate the function at specific critical points, such as when \( x \) makes the argument \( \frac{\pi}{2} \) for sine. Why? Because sine reaches its maximum value of 1 at \( \frac{\pi}{2} \), this makes calculations straightforward.

• For \( x = 1 \), we computed \( \sin \left( \frac{\pi \cdot 1}{2} \right) = \sin \frac{\pi}{2} = 1 \). This simplifies the contribution of the trigonometric part to the limit.

Thus, understanding the specific behavior of the sine function is crucial to solving such limit problems.

Graphical Analysis

Graphical analysis offers a visual perspective to understand limits and can confirm analytical solutions. Graphing the function \( e^x \sin \frac{\pi x}{2} \) is a practical method to verify predictions about its behavior as \( x \) approaches a certain value, like 1.

A graph reveals the interaction between the exponential and trigonometric parts of the function. Around \( x = 1 \), the graph should visibly approach the value derived analytically, here \( e \). If plotted accurately, the function ought to illustrate that as \( x \to 1 \), the function curves consistently towards \( y = e \).

• This confirmation via graphical analysis ensures that our calculation of \( \lim _{x \rightarrow 1} e^x \sin \frac{\pi x}{2} = e \) was indeed correct.
• Beyond confirming, this visual tool can illuminate nuances, such as how close intervals around \( x = 1 \) behave in approximating the limit.

Approaching limits graphically complements the mathematical tools students use, providing a fuller picture of how functions behave.