Problem 12 - Calculus for Biology and Medicine

Question

In Problems 1-32, use a table or a graph to investigate each limit. $$ \lim _{t \rightarrow e} \ln \left(\frac{1}{t}\right) $$

Step-by-Step Solution

Answer

The limit is -1.

Step 1: Recognize the Limit Expression

We need to find the limit as \( t \) approaches \( e \) of the expression \( \ln \left( \frac{1}{t} \right) \). This involves both the natural logarithm function and a rational function.

Step 2: Simplify the Expression

Rewrite \( \ln \left( \frac{1}{t} \right) \) using the logarithmic identity \( \ln \left( \frac{a}{b} \right) = \ln(a) - \ln(b) \). In this case, it's simply \( \ln(1) - \ln(t) = 0 - \ln(t) = -\ln(t) \).

Step 3: Substitute \( t = e \)

Now substitute \( t = e \) into the simplified expression. We get \( -\ln(e) \).

Step 4: Evaluate the Logarithmic Function

Since \( \ln(e) = 1 \) (because \( e \) is the base of the natural logarithm), we can substitute this into our expression to get \(-1\).

Step 5: State the Limit

The limit is \(-1\). Thus, \( \lim _{t \rightarrow e} \ln \left( \frac{1}{t} \right) = -1 \).

Key Concepts

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is a fundamental concept in calculus. It is the logarithm to the base \( e \), where \( e \) is approximately equal to 2.71828. The natural logarithm has unique properties that make it particularly useful in mathematics. One important property is its inverse relation to the exponential function. This means that \( \ln(e^x) = x \) and \( e^{\ln(x)} = x \). Therefore, \( \ln(e) = 1 \) because the exponent that \( e \) must be raised to in order to produce \( e \) is 1.

The natural logarithm is widely used for simplifying expressions and solving equations involving exponential growth or decay. For example, in calculus, it's often utilized to solve integrals and limits, such as the exercise given where equivalences like \( \ln \left(\frac{1}{t}\right) \) arise. Understanding how to manipulate expressions using \( \ln(x) \) can clarify the process of finding limits.

Rational Functions

Rational functions are another key component in the given limit problem. A rational function is defined as the ratio of two polynomials, such as \( \frac{1}{t} \) in this exercise. These functions are interesting because their properties are guided by the degrees of the numerator and the denominator.

When approaching limits involving rational functions, particularly as the variable reaches infinity or a certain point, specific behaviors can be observed. For example, \( \frac{1}{t} \) represents a function that approaches zero as \( t \) becomes very large. As \( t \) approaches a particular point, like \( e \), rational functions may lead to direct substitutions if the function is defined or may require simplification, such as using logarithm properties.

Understanding rational functions is crucial for analyzing complex relationships in calculus, especially when combined with other functions like logarithms.

Logarithmic Identities

To solve limits effectively involving logarithms, an understanding of logarithmic identities is essential. These identities help simplify complex expressions into more manageable forms. One of the most utilized identities is \( \ln(ab) = \ln(a) + \ln(b) \), which helps break down products inside a logarithm into sums. Another key identity, as used in the problem solution, is \( \ln \left( \frac{a}{b} \right) = \ln(a) - \ln(b) \).

In the given exercise, this second identity was pivotal. Applying this to \( \ln \left( \frac{1}{t} \right) \), the expression changes to \( \ln(1) - \ln(t) = 0 - \ln(t) = -\ln(t) \). These identities enable us to transform complicated logarithmic expressions into simpler terms, making evaluations much more straightforward.

Mastering these identities broadens mathematical problem-solving techniques and assists in navigating through otherwise challenging calculus problems.

Evaluating Limits

Evaluating limits is a central concept in calculus, providing insights into the behavior of functions as variables approach specific points or infinity. The process often involves simplification of the original expression and substitution of the limiting value.

In this exercise, evaluating the limit \( \lim _{t \rightarrow e} \ln \left( \frac{1}{t} \right) \) required transforming the expression using logarithmic identities. After simplifying to \(-\ln(t)\), and substituting \( t = e \), the problem reduces to evaluating \(-\ln(e)\), which equals \(-1\) because \( \ln(e) = 1 \). Hence, the limit is found to be \(-1\).

Limits play an essential role in understanding continuity, derivatives, and integrals, helping to describe how functions behave at critical points without graphing every scenario. Familiarity with evaluating limits enables deeper comprehension of how functions interact and aids in solving vast arrays of calculus problems.