Problem 54 - Calculus

Question

Compute \(\lim _{x \rightarrow 0} \frac{b^{x}-1}{x}\)

Step-by-Step Solution

Answer

The limit is \(\ln b\).

Step 1: Identify the Limit Problem

We are given a limit problem involving exponential function, \[ \lim_{x \to 0} \frac{b^x - 1}{x} \]which requires us to evaluate the expression as \(x\) approaches 0.

Step 2: Recognize L'Hôpital's Rule Applicability

The expression is in an indeterminate form \(\frac{0}{0}\) as \(x\) approaches 0. Thus, we can apply L'Hôpital's Rule, which involves taking the derivative of the numerator and the derivative of the denominator separately.

Step 3: Differentiate the Numerator and Denominator

Differentiate the numerator and denominator with respect to \(x\):- Derivative of the numerator, \(b^x - 1\), is \(b^x \ln b\).- Derivative of the denominator, \(x\), is 1.

Step 4: Apply L'Hôpital's Rule

By applying L'Hôpital's Rule, we can rewrite the limit as:\[ \lim_{x \to 0} \frac{b^x \ln b}{1} = \lim_{x \to 0} b^x \ln b \]

Step 5: Evaluate the Simplified Limit

As \(x\) approaches 0, \(b^x\) approaches \(b^0 = 1\). So the limit simplifies to:\[ \ln b \times 1 = \ln b \]

Step 6: Conclude the Result

After applying and simplifying, the expression evaluates to \(\ln b\) as \(x\) approaches 0. Therefore, the final limit is \(\ln b\).

Key Concepts

L'Hôpital's Rule

L'Hôpital's Rule is an essential technique in calculus that helps us evaluate limits, especially when dealing with indeterminate forms like \(\frac{0}{0}\) and \(\frac{\infty}{\infty}\). These forms arise when the direct substitution of a limit yields undefined mathematical expressions.

To apply L'Hôpital's Rule, remember these steps:

• Ensure that the limit operation results in an indeterminate form.
• Take the derivative of the numerator and the derivative of the denominator separately.
• Re-evaluate the limit with these new derivatives.

For example, in the problem \(\lim_{x \to 0} \frac{b^x - 1}{x}\), by directly substituting \(x = 0\), we initially get the form \(\frac{0}{0}\). This is a perfect candidate for L'Hôpital's Rule, which allows us to differentiate the components to resolve the limit.

Exponential Functions

Exponential functions are powerful mathematical tools often represented in the form \(b^x\), where \(b\) is a positive real number base, and \(x\) is the exponent. A critical feature of exponential functions is their growth behavior. They increase rapidly as the exponent increases, especially when the base \(b\) is greater than 1.

One of the key properties of exponential functions is that they can exhibit different behaviors as the exponent \(x\) approaches certain values. For instance, as \(x\) approaches 0, the expression \(b^x\) evaluates to 1, since any nonzero number to the power of 0 equals 1.

• The derivative of an exponential function with respect to \(x\) is \(b^x \ln b\). This property is fundamental in problems involving exponential limits and helps significantly in applying L'Hôpital's Rule to indeterminate forms involving exponentials.

Indeterminate Forms

Indeterminate forms are expressions that do not have a clear or immediate outcome or value. Common indeterminate forms include \(\frac{0}{0}\), \(\frac{\infty}{\infty}\), \(0 \times \infty\), among others.

Indeterminate forms suggest that more information is needed to evaluate the expression, such as through algebraic manipulation or the usage of calculus techniques like limits or derivatives.

In the given exercise, when attempting to compute \(\lim_{x \to 0} \frac{b^x - 1}{x}\), the direct substitution of \(x = 0\) leads to the indeterminate form \(\frac{0}{0}\). This indicates that further analysis, such as employing L'Hôpital's Rule, is required to find the limit accurately.

• Recognizing indeterminate forms is crucial in calculus, triggering deeper analysis to solve limit problems effectively.

Derivatives

Derivatives are a cornerstone of calculus and serve as a measure of how a function changes as its input changes. They represent the function's instantaneous rate of change or the slope of the tangent line at any point along the curve.

In limit problems, derivatives are especially useful when applying L'Hôpital's Rule to resolve indeterminate forms.

To find the derivative of a function, use rules such as the power rule, the product rule, or the chain rule, depending on the function's complexity.

• For the function \(b^x\), its derivative is calculated as \(b^x \ln b\), utilizing the property of exponential functions.
• The derivative of a linear function like \(x\) is simply 1.

In the problem \(\lim_{x \to 0} \frac{b^x - 1}{x}\), taking the derivative of both the numerator \(b^x - 1\) and denominator \(x\) allowed transformation from an indeterminate form to a limit that simplifies directly to a numerical answer.