Problems Based on Exponential Limits

Understanding exponential limits is crucial for students studying calculus, as they often appear in various mathematical contexts, such as growth and decay processes, compound interest, and in the analysis of algorithms. In this guide, we will explore the concept of exponential limits, their properties, and how to solve problems involving them.

Exponential Functions

An exponential function is a mathematical function of the form:

$$ f(x) = a \cdot b^{x} $$

where:

• $a$ is a constant,
• $b$ is the base of the exponential (with $b > 0$ and $b \neq 1$),
• $x$ is the exponent.

Limits Involving Exponential Functions

When dealing with limits of exponential functions, we often encounter the following types:

1. $\lim_{x \to \infty} a \cdot b^{x}$
2. $\lim_{x \to -\infty} a \cdot b^{x}$
3. $\lim_{x \to c} a \cdot b^{x}$, where $c$ is a finite number.

The behavior of these limits depends on the base $b$ of the exponential function.

Table of Exponential Limit Behaviors

Base $b$	$\lim_{x \to \infty} b^{x}$	$\lim_{x \to -\infty} b^{x}$	$\lim_{x \to c} b^{x}$
$b > 1$	$\infty$	$0$	$b^{c}$
$0 < b < 1$	$0$	$\infty$	$b^{c}$

Important Points

• When $b > 1$, the function grows without bound as $x$ approaches infinity, and approaches zero as $x$ approaches negative infinity.
• When $0 < b < 1$, the function approaches zero as $x$ approaches infinity, and grows without bound as $x$ approaches negative infinity.
• For any base $b$ (except $b = 1$), as $x$ approaches a finite number $c$, the limit is simply the function value at $c$, which is $b^{c}$.

Formulas Involving Exponential Limits

One of the most important exponential limits to remember is:

$$ \lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^{x} = e $$

where $e$ is Euler's number, approximately equal to 2.71828.

Another useful limit is:

$$ \lim_{x \to 0} \frac{e^{x} - 1}{x} = 1 $$

This limit is often used in the derivation of the derivative of the exponential function.

Examples

Example 1: Basic Exponential Limit

Calculate the limit:

$$ \lim_{x \to \infty} 2^{x} $$

Solution:

Since the base $b = 2$ is greater than 1, the limit as $x$ approaches infinity is:

$$ \lim_{x \to \infty} 2^{x} = \infty $$

Example 2: Exponential Limit with a Negative Exponent

Calculate the limit:

$$ \lim_{x \to -\infty} 3^{-x} $$

Solution:

First, rewrite the function as:

$$ 3^{-x} = \frac{1}{3^{x}} $$

Since the base $b = 3$ is greater than 1, the limit as $x$ approaches negative infinity is:

$$ \lim_{x \to -\infty} \frac{1}{3^{x}} = \lim_{x \to -\infty} \frac{1}{\infty} = 0 $$

Example 3: Exponential Limit at a Finite Point

Calculate the limit:

$$ \lim_{x \to 2} 5^{x} $$

Solution:

Since $x$ approaches a finite number, the limit is simply the function value at that point:

$$ \lim_{x \to 2} 5^{x} = 5^{2} = 25 $$

Example 4: Exponential Limit Involving Euler's Number

Calculate the limit:

$$ \lim_{x \to 0} \frac{e^{x} - 1}{x} $$

Solution:

This is a standard limit that is known to be equal to 1:

$$ \lim_{x \to 0} \frac{e^{x} - 1}{x} = 1 $$

Understanding these concepts and practicing various problems will help students become proficient in solving problems based on exponential limits. It is important to remember the behavior of exponential functions as the exponent approaches infinity, negative infinity, or a finite number, and to be familiar with the key limits involving Euler's number.