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Chapter 4: Problem 12

Complete the statements for \(f(x)=a^{-x}+c\) with \(a>1\). (a) As \(x \rightarrow \infty, f(x) \rightarrow\)_____. (b) As \(x \rightarrow-\infty, f(x) \rightarrow\)_____.

Short Answer

Expert verified
(a) \(c\); (b) \(+\infty\)."

Step by step solution

01

Understand the Behavior of the Function

The function given is \(f(x) = a^{-x} + c\). Here, \(a > 1\), so \(a^{-x}\) is an exponential decay function as \(x\) increases or decreases.
02

Analyze the Limit as \(x \to \infty\)

As \(x\) approaches infinity, the power \(-x\) becomes a large negative number. This means \(a^{-x}\), which is equivalent to \(\frac{1}{a^x}\), becomes very small and approaches zero. Thus, the function \(f(x)\) approaches \(c\).
03

Analyze the Limit as \(x \to -\infty\)

As \(x\) approaches negative infinity, \(-x\) becomes a large positive number. Therefore, \(a^{-x} = a^{|x|}\) becomes a large positive number as \(|x|\) grows infinitely large. Hence, \(f(x)\) approaches infinity (\(+\infty\)).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Asymptotic Behavior
Asymptotic behavior refers to the way a function behaves as the input approaches a particular value, often infinity. In this context, we examine how the function \( f(x) = a^{-x} + c \) behaves as \( x \) becomes extremely large in either the positive or negative direction. This concept is crucial when analyzing functions to understand long-term trends beyond the immediate polygonal line defined by the function.

For \( f(x) = a^{-x} + c \):
  • As \( x \rightarrow \infty \): Since \( a^{-x} \) becomes very small, the function \( f(x) \) approaches the constant \( c \). Therefore, the function's asymptote is a horizontal line at \( y = c \).
  • As \( x \rightarrow -\infty \): The term \( a^{-x} \) grows very large, causing \( f(x) \) to increase without bound. Thus, it approaches positive infinity, showing a vertical asymptotic behavior.
Asymptotic behavior is integral in understanding functions with complex terms by focusing on behavior at extreme values instead of middle-range operations.
Limits at Infinity
Limits at infinity examine the value a function approaches as \( x \) approaches \( \pm \infty \). It is often used to describe the end behavior of polynomial, rational, exponential, and trigonometric functions. Understanding these limits is essential in calculus to predict and describe trends in data or natural phenomena.

For the function \( f(x) = a^{-x} + c \):
  • As \( x \rightarrow \infty \): The power \( -x \) becomes a huge negative value, making \( a^{-x} \) approach zero. Consequently, \( f(x) \) tends towards \( c \), representing the horizontal limit at infinity.
  • As \( x \rightarrow -\infty \): The power \( -x \) becomes a large positive value, causing \( a^{-x} \) to increase significantly. Thus, \( f(x) \) approaches \( +\infty \), indicating that no finite limit can hold as \( x \rightarrow -\infty \).
These calculations clarify how \( f(x) \) behaves in extreme scenarios, showing that one direction stabilizes as the function aligns with the constant \( c \), while the other rises unabated.
Negative Exponents
Negative exponents are essential in understanding phenomena where growth or decay is captured succinctly. When a base \( a > 1 \) is raised to a negative exponent, the expression \( a^{-x} \) is equivalent to \( \frac{1}{a^x} \). This feature is foundational in describing how quantities decrease exponentially.

In the function \( f(x) = a^{-x} + c \):
  • For positive \( x \), \( a^{-x} \) represents a decline or decay, because each increase in \( x \) results in a smaller value, approaching zero as \( x \rightarrow \infty \).
  • For negative \( x \), the value \( -x \) itself is positive, making \( a^{-x} \), or \( a^{|x|} \), a representation of exponential growth rather than decay.
Understanding negative exponents aids in demystifying complex behavior and builds a bridge between exponential growth and decay distinctions.

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