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.
$f (x) = - 3 e^{- x}$

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Identify the function: $ f(x) = -3e^{-x} $. This is an exponential function with a negative exponent.

Analyze the limit as $ x \to \infty $: Since $ e^{-x} \to 0 $ as $ x \to \infty $, $ f(x) = -3e^{-x} \to 0 $. Therefore, the function approaches 0 from below.

Analyze the limit as $ x \to -\infty $: As $ x \to -\infty $, $ e^{-x} \to \infty $, so $ f(x) = -3e^{-x} \to -\infty $.

Determine horizontal asymptote: From the limit as $ x \to \infty $, the horizontal asymptote is $ y = 0 $.

Sketch the graph: The graph approaches the horizontal asymptote $ y = 0 $ from below as $ x \to \infty $ and decreases without bound as $ x \to -\infty $.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

End Behavior of Functions

End behavior refers to the behavior of a function as the input values approach positive or negative infinity. For transcendental functions like exponential functions, understanding end behavior helps predict how the function behaves far away from the origin. This is crucial for sketching graphs and identifying asymptotes.

Limits

Limits are fundamental in calculus, used to describe the value that a function approaches as the input approaches a certain point. In the context of end behavior, limits at infinity help determine the horizontal asymptotes of a function. For the function f(x) = -3e^(-x), evaluating the limit as x approaches infinity reveals its behavior.

Asymptotes

Asymptotes are lines that a graph approaches but never touches. They can be vertical, horizontal, or oblique. For the function f(x) = -3e^(-x), identifying horizontal asymptotes involves analyzing the limits at infinity, which indicates the value the function approaches as x becomes very large.

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Textbook Question

Determine the following limits.

$\lim_{θ \to 0^{-}} \frac{\sin θ}{\cos^{2} θ - 1}$

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Textbook Question

Determine $$\lim$_{x$\rightarrow\]\infty$}f$\left$(x$\right$)$ and $$\lim$_{x$\rightarrow$-$\infty$}f$\left$(x$\right$)$ for the following functions. Then give the horizontal asymptotes of $f$ (if any).