Problem 79 - Calculus - AP Edition

Question

Find the limit by interpreting the expression as an appropriate derivative. $$\lim _{h \rightarrow 0} \frac{10^{h}-1}{h}$$

Step-by-Step Solution

Answer

The limit is \( \ln(10) \).

Step 1: Identify Similarity to Derivative Definition

The expression \( \lim _{h \rightarrow 0} \frac{10^{h}-1}{h} \) resembles the definition of the derivative of a function at a point, which is \( \lim_{h\to 0}\frac{f(a+h)-f(a)}{h} \). Here, we note that it resembles the derivative of the function \( f(x) = a^x \) at \( x = 0 \) and \( a = 10 \).

Step 2: Recall Derivative of an Exponential Function

The derivative of the exponential function \( f(x) = a^x \) at any point is given by the formula \( f'(x) = a^x \ln(a) \). For \( x = 0 \), it simplifies to \( f'(0) = a^0 \ln(a) = \ln(a) \), since \( a^0 = 1 \).

Step 3: Apply the Derivative to Solve Limit

By recognizing the function \( f(x) = 10^x \) and noting the form of the limit as corresponding to the derivative at \( x = 0 \), we deduce that the limit is the derivative \( f'(0) = \ln(10) \), as calculated from \( a = 10 \).

Step 4: Conclusion

The limit \( \lim _{h \rightarrow 0} \frac{10^{h}-1}{h} \) is the derivative of the function \( 10^x \) at \( x = 0 \), which is \( \ln(10) \).

Key Concepts

Derivative Definition

The derivative of a function is a fundamental concept in calculus that describes how a function changes at any given point. While it may sound complex, think of a derivative as a tool that helps us understand how steep a graph is at any point, or simply how the function's value increases or decreases. The formal definition of a derivative can be understood as the limit of the average rate of change of the function over a very small interval. In mathematical terms, if you want to find the derivative of a function \( f(x) \) at a point \( a \), you would use the formula:

• \( f'(a) = \lim_{h\to 0} \frac{f(a+h) - f(a)}{h} \)

This expression may look daunting, but it's essentially measuring how much the function value changes per unit change in \( x \), as the change in \( x \) becomes infinitesimally small.
Your goal with derivatives is to grasp how tiny changes in \( x \) result in changes in \( f(x) \). It's like zooming in really close to a graph at point \( a \) until it looks like a straight line, letting you find its slope.

Exponential Function Derivative

Exponential functions are powerful in mathematics due to their unique growth properties. Knowing how to differentiate them is crucial because they appear in many real-life scenarios, like population growth or radioactive decay. When dealing with exponential functions of the form \( f(x) = a^x \), we have a special rule for their derivatives:

• \( f'(x) = a^x \ln(a) \)

This formula means that the derivative of an exponential function is a multiple of the original function itself. The multiplying factor \( \ln(a) \) is the natural logarithm of the base \( a \). It's essential because it transforms our exponential base into a continuous growth rate.
When we have the function \( 10^x \), its derivative at any point \( x \) is \( 10^x \ln(10) \). At \( x = 0 \), this simplifies further because \( 10^0 = 1 \), leading to the derivative being simply \( \ln(10) \). This simplification is key for solving limit problems involving exponential functions, as seen in the original exercise's solution.

Limit Interpretation

Understanding the concept of limits is crucial to mastering calculus. Limits help us work with values that seem to approach a number, without ever reaching it directly. In the context of derivatives, limits enable us to define how functions behave at certain points, even when the behaviour isn't straightforward.
The expression \( \lim_{h \rightarrow 0} \frac{10^{h}-1}{h} \) we encounter in the exercise is interpreted as a derivative limit form. It's analogous to the definition of a derivative, which makes the calculation of such limits manageable. By realizing that the limit resembles the derivative \( f'(0) \) of \( 10^x \) at \( x = 0 \), we can solve it using knowledge of exponential derivatives.

• The key takeaway is that the limit equates to \( \ln(10) \), which would be the derivative of \( 10^x \) at \( x = 0 \).

With limits, you're often examining the behaviour as you approach a value rather than substituting directly. Recognizing forms that match known derivatives can greatly simplify the process of solving these problems.