Question

Determine the end behavior of the following transcendental functions by analyzing appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist. $$f(x)=-3 e^{-x}$$

Short answer

Answer: The end behavior of the function $$f(x) = -3e^{-x}$$ is such that, as x approaches negative infinity, the function goes to negative infinity, and as x approaches positive infinity, the function approaches 0. The asymptote for this function is a horizontal asymptote at y = 0.

Step-by-step solution

End Behavior as x Approaches Negative Infinity

To find the end behavior as x approaches negative infinity, we need to analyze the limit as x approaches negative infinity for the function $$f(x) = -3e^{-x}$$. To find this limit, we have: $$\lim_{x \to -\infty} -3e^{-x}$$ As x goes to negative infinity, the exponent $$-x$$ goes to positive infinity. Therefore, we have: $$\lim_{x \to -\infty} -3e^{-x} = -3e^{\infty} = -\infty$$ This means that as x approaches negative infinity, the function goes to negative infinity.

End Behavior as x Approaches Positive Infinity

Similarly, we will now analyze the limit as x approaches positive infinity for the function $$f(x) = -3e^{-x}$$. To find this limit, we have: $$\lim_{x \to \infty} -3e^{-x}$$ As x goes to positive infinity, the exponent $$-x$$ goes to negative infinity. Therefore, we have: $$\lim_{x \to \infty} -3e^{-x} = -3e^{-\infty} = -3(0) = 0$$ This means that as x approaches positive infinity, the function approaches 0.

Identifying Asymptotes

In this case, we can see that the function approaches 0 as x approaches positive infinity. This suggests that there is a horizontal asymptote at y=0. We can write the equation of this horizontal asymptote as: $$y = 0$$ There are no vertical asymptotes in this function since the exponential function is well-defined for all real x values.

Sketching the Graph

Now that we have the end behavior and asymptotes, we can sketch a simple graph of the function. From the previous steps, we know the graph goes to negative infinity as x approaches negative infinity and approaches 0 as x approaches positive infinity. Also, there's a horizontal asymptote at y=0. Using this information, the graph should show a curve that starts at $$y = -\infty$$ when x approaches negative infinity, gets closer to y=0 as x increases, and approaches the horizontal asymptote at y=0 as x approaches positive infinity.

Key concepts

Limits of Exponential Functions

Understanding the behavior of exponential functions as values approach infinity is essential for mastering calculus concepts. One commonly studied transcendental function is the exponential function, typically represented in the form of \( f(x) = ae^{bx} \), where \( a \) and \( b \) are constants. To determine end behavior, we look at the limits of the function as \( x \) approaches positive or negative infinity.

For the function \( f(x) = -3e^{-x} \), as \( x \) approaches negative infinity, the limit is \( -3e^{\infty} \), which diverges to negative infinity. Conversely, as \( x \) approaches positive infinity, the negative exponent makes the term \( e^{-x} \) shrink towards zero, leading to a limit of zero.

The ability to analyze these limits provides insight into the function's growth and decay trends, which are pivotal to understanding its long-term behavior.

Sketching Graphs of Functions

When you're tasked with sketching graphs, it's crucial to identify key characteristics—like intercepts, end behavior, and particularly, asymptotes. The graph of any function is a visual representation that serves to illuminate these elements. For the function \( f(x) = -3e^{-x} \), the sketching process begins by noting its end behavior—plummeting towards negative infinity as \( x \) goes to negative infinity, and leveling off towards zero as \( x \) goes to positive infinity.

An effective sketch shows these tendencies: starting from the left, the curve descends steeply, gradually flattening out as it approaches the horizontal asymptote at y=0. Overall, sketching the graph of a function like this serves as a vital step in confirming our calculations and understanding the function's overall shape and behavior.

Horizontal Asymptotes

Horizontal asymptotes give us information about the behavior of a function as \( x \) stretches towards infinity in either direction. They are the lines that the graph of the function approaches but never actually touches. The presence of a horizontal asymptote answers the question, 'What value does the function get arbitrarily close to, but never actually reach, as x goes to infinity or negative infinity?'.

For our function \( f(x) = -3e^{-x} \), the horizontal asymptote is at \( y = 0 \). This horizontal asymptote signifies that as \( x \) increases without bound, the function values approach zero, but never surpass that threshold. It's important to note that not all functions have horizontal asymptotes, and some might have more than one, depending on their behavior at infinity. Recognizing these asymptotes is fundamental to predicting the function's long-range behavior and is a key tool when sketching the function's graph.