Problem 33 - Essential Calculus

Question

In Exercises \(15-36,\) find the limit. $$ \lim _{x \rightarrow \infty} \log _{10}\left(1+10^{-x}\right) $$

Step-by-Step Solution

Answer

The solution to the limit problem \(\lim _{x \rightarrow \infty} \log _{10}\left(1+10^{-x}\right)\) is \(0\).

Step 1: Set up the Limit Problem

To start, let's write out the limit problem: \[\lim _{x \rightarrow \infty} \log _{10}\left(1+10^{-x}\right)\]

Step 2: Apply Infinite Limit Properties

When \(x\) tends to infinity, \(10^{-x}\) will tend to zero as \(10^{-\infty}= 0\). Then, the function becomes \[\lim _{x \rightarrow \infty} \log _{10}(1 + 0)\]

Step 3: Simplify the Logarithmic Function

Simplify the function to be \[\lim _{x \rightarrow \infty} \log _{10}(1) \]Here, using the property of logarithm, \(\log_{a}(b) = 0\), if \(b = 1\). Hence, \(\lim _{x \rightarrow \infty} \log _{10}(1) = 0\).

Key Concepts

Logarithmic Functions

In mathematics, logarithmic functions allow us to solve equations involving exponents more easily. Essentially, a logarithm is the inverse operation to exponentiation.
For instance, if you know the equation \[ a^b = c \], then the logarithm can help you find the value of \( b \) by rewriting it as \( b = \log_a(c) \). This means "the power to which the base \( a \) must be raised to obtain \( c \)."

The most common types of logarithms are:

• Common logarithms (\( \log_{10} \)),
• Natural logarithms (\( \ln \)),
• Binary logarithms (\( \log_2 \)).

Each type uses a different base but functions in the same way to simplify complex exponential expressions.
Understanding logarithms is essential for solving many calculus problems, particularly those involving limits.

Infinite Limits

The concept of infinite limits is crucial in calculus. It refers to the behavior of a function as the variable approaches infinity.
In simpler terms, you are analyzing what happens to the function when the input grows larger and larger without bound.

For example, consider the expression \( x \rightarrow \infty \). This notation means that \( x \) grows very large.
If you have \( 10^{-x} \), as \( x \) approaches infinity, \( 10^{-x} \) gets closer and closer to zero. The idea is to determine the behavior of the function based on this change.

In the original exercise, as \( x \rightarrow \infty \), \( 1 + 10^{-x} \) approaches 1, making it easier to simplify the final limit expression.

Properties of Logarithms

Understanding the properties of logarithms makes it easier to work through expressions and limits involving logarithmic functions. These properties help simplify calculations, especially when dealing with complex expressions.
Here are some fundamental properties:

• \( \log_{b}(1) = 0 \) means the logarithm of 1 in any base is zero, as any number raised to the power of zero is 1.
• \( \log_{b}(b) = 1 \) indicates that the logarithm of a number in its own base is one.
• Product Rule:\( \log_{b}(xy) = \log_{b}(x) + \log_{b}(y) \)
• Quotient Rule:\( \log_{b}(\frac{x}{y}) = \log_{b}(x) - \log_{b}(y) \)
• Power Rule:\( \log_{b}(x^y) = y \cdot \log_{b}(x) \)

In the initial solution, we leveraged the property \( \log_{b}(1) = 0 \) to conclude that the logarithm of 1 in any base, as the limit approaches infinity, is zero.