Problem 14 - APEX Calculus

Question

Evaluate the given limits using the graph of the function. \(f(x)=2^{x}+10\) (a) \(\lim _{x \rightarrow-\infty} f(x)\) (b) \(\lim _{x \rightarrow \infty} f(x)\)

Step-by-Step Solution

Answer

(a) 10, (b) \(\infty\)

Step 1: Understand the Problem

We need to evaluate the limits of the function \(f(x) = 2^x + 10\) as \(x\) approaches \(-\infty\) and \(\infty\). This means examining the behavior of \(f(x)\) as \(x\) becomes very large in either the positive or negative direction.

Step 2: Analyze the Function for \(x \rightarrow -\infty\)

Consider the term \(2^x\). As \(x\) approaches \(-\infty\), \(2^x\) approaches \(0\) because the exponential function approaches zero for very large negative values of \(x\). Thus, \(f(x) = 2^x + 10\) approaches \(10\) as \(x\) approaches \(-\infty\).

Step 3: Compute \(\lim _{x \rightarrow -\infty} f(x)\)

Since \(2^x\) approaches \(0\) as \(x\) approaches \(-\infty\), we find that \(\lim _{x \rightarrow -\infty} f(x) = 10\).

Step 4: Analyze the Function for \(x \rightarrow \infty\)

For \(x\) approaching \(\infty\), \(2^x\) becomes very large because it is an exponential growth function. The function \(f(x) = 2^x + 10\) will also tend towards infinity as \(x\) approaches \(\infty\).

Step 5: Compute \(\lim _{x \rightarrow \infty} f(x)\)

Since \(2^x\) becomes infinitely large as \(x\) goes to \(\infty\), \(\lim _{x \rightarrow \infty} f(x) = \infty\).

Key Concepts

Exponential Function

An exponential function is one of the most important classes of mathematical functions, defined generally as a function in the form of \( f(x) = a^x \), where \( a \) is a positive constant. This type of function exhibits a rate of change that is directly proportional to its current value, making it grow very quickly. In our specific problem with \( f(x) = 2^x + 10 \), the base of the exponential, \( 2 \), indicates that the function doubles as \( x \) increases by 1.

Exponential functions have distinct characteristics:

• They increase rapidly, reflecting exponential growth when the base \( a \) is greater than 1.
• They approach zero asymptotically as \( x \) gets very large in the negative direction when \( a \) is greater than 1.

The key insight for limits is observing the behavior of the exponential component. As \( x \) becomes very large positively, \( 2^x \) grows without bounds. Conversely, when \( x \) becomes very negative, \( 2^x \) diminishes towards zero.

Infinity

In mathematics, infinity denotes a concept regarding boundlessness. It isn't a number but an idea of something that continues endlessly. When dealing with limits of a function, considering \( x \) approaching infinity means observing the function's behavior as the input grows indefinitely large. For example, \( \lim_{x \to \infty} f(x) \) checks what value \( f(x) \) approaches as \( x \) becomes infinitely big.

When evaluating limits of function like \( f(x) = 2^x + 10 \):

• As \( x \rightarrow \infty \), the argument of the \( 2^x \) part grows without bound, causing the entire function to tend to infinity as well.
• As \( x \rightarrow -\infty \), the exponential part \( 2^x \) approaches zero, simplifying the analysis to focus on the constant part, which is 10.

This reflects how functions behave as \( x \) tends to very large positive or negative values, revealing insights into their long-term behavior.

Asymptotic Behavior

Asymptotic behavior describes how a function behaves as the input approaches a particular value, often infinity or negative infinity. It supplies crucial insights into the end behavior of functions.

In our function \( f(x) = 2^x + 10 \):

• For \( x \rightarrow -\infty \), \( 2^x \) heads towards zero, causing \( f(x) \) to asymptotically approach the horizontal line \( y = 10 \). The function gets closer and closer to 10 but never really gets there, staying asymptotically attached.
• For \( x \rightarrow \infty \), \( 2^x \) makes \( f(x) \) rise infinitely. Thus, the function doesn't have a limiting horizontal asymptote in this direction, as it goes upwards indefinitely.

Understanding asymptotic behavior is fundamental in predicting how functions behave as they move towards extreme values of \( x \). It helps predict a function's end behavior without graphically plotting every point.