Problem 35 - Calculus Early Transcendentals

Question

Calculator limits Estimate the value of the following limits by creating a table of function values for \(h=0.01,0.001,\) and 0.0001 and \(h=-0.01,-0.001,\) and -0.0001. $$\lim _{h \rightarrow 0} \frac{\ln (1+h)}{h}$$

Step-by-Step Solution

Answer

Based on the table created and the evaluation of the function values as h tends to 0, estimate the value of the limit: $$\lim_{h \rightarrow 0} \frac{\ln (1+h)}{h}$$

Step 1: Function Definition

First, define the function: $$f(h) = \frac{\ln(1 + h)}{h}$$

Step 2: Creating the Table

Now, we will create a table of function values for the given values of h: \(h = 0.01, 0.001, 0.0001, -0.01, -0.001,\) and \(-0.0001\). | h | f(h) | |---|--------| | 0.01 | f(0.01) | | 0.001 | f(0.001) | | 0.0001 | f(0.0001) | | -0.01 | f(-0.01) | | -0.001 | f(-0.001) | | -0.0001 | f(-0.0001) |

Step 3: Calculate Function Values

Next, we will calculate the function values for each entry in the table by plugging in each h value: | h | f(h) | |---|--------| | 0.01 | \(\frac{\ln(1.01)}{0.01}\) | | 0.001 | \(\frac{\ln(1.001)}{0.001}\) | | 0.0001 | \(\frac{\ln(1.0001)}{0.0001}\) | | -0.01 | \(\frac{\ln(0.99)}{-0.01}\) | | -0.001 | \(\frac{\ln(0.999)}{-0.001}\) | | -0.0001 | \(\frac{\ln(0.9999)}{-0.0001}\) |

Step 4: Evaluate Function Values

Now, we will evaluate the function values using a calculator or computational software: | h | f(h) | |---|--------| | 0.01 | 0.9967 | | 0.001 | 0.9997 | | 0.0001 | 0.99997 | | -0.01 | 1.0034 | | -0.001 | 1.0003 | | -0.0001 | 1.00003 |

Step 5: Estimate the Limit

Lastly, we will observe the behavior of the function values as h tends to 0. From the table, we can see that the function values are getting closer to 1 as h approaches 0. Therefore, we can estimate the limit as follows: $$\lim_{h \rightarrow 0} \frac{\ln (1+h)}{h} \approx 1$$

Key Concepts

Logarithms

Logarithms are a fundamental concept in mathematics, particularly in calculus. They allow us to solve equations involving exponential functions. The logarithm we often deal with here is the natural logarithm (ln), which is the inverse of the exponential function with base e, where e is approximately 2.718. Simply put, the natural logarithm of a number is the power to which e must be raised to get that number. Understanding how logarithms work can simplify calculations and enable us to grasp how functions behave. When dealing with limits, expressions like \( \ln (1 + h) \) become crucial, especially as h approaches zero. The natural logarithm of a number close to 1 has special properties that make it predictable, supporting our exploration of limits.

Approaching Zero

The concept of a number approaching zero is essential in understanding limits in calculus. When we say a variable \( h \) approaches zero, we are interested in how a function behaves nearby this point, rather than at it directly. In practical terms, we evaluate the function for values of \( h \) that are very small in absolute terms, both positive and negative, to observe the trend as \( h \) gets closer to zero.

• Consider how the values change in the table for positive and negative \( h \).
• This observation helps to anticipate how the function settles into a particular value, here estimated to be 1.

As \( h \) approaches zero in this limit exercise, the incremental change \( \ln (1 + h) / h \) becomes smaller and stabilizes, crucial for understanding calculus-based problems.

Function Evaluation

Function evaluation involves computing the output value of a function for a particular input, which is vital for analyzing limits. In this context, it's about calculating \( f(h) = \frac{\ln(1 + h)}{h} \) for several values of \( h \) close to zero. Function evaluation provides insight into how a function behaves in the vicinity of a particular point.

• Using a calculator, you compute values at given points like \( 0.01 \) or \( -0.01 \).
• Repeated evaluations for smaller and smaller values teach us about the trend and limit.

The goal of function evaluation is to see if the outputs reveal a pattern or "limit" as \( h \) approaches zero. Here, outputs are converging towards 1, thus helping determine the limit. This approach is not only methodical but essential for accurate predictions and deeper understanding in calculus.