Problem 55 - A Graphical Approach to Precalculus with Limits

Question

Evaluate each limit. (a) \(\lim _{x \rightarrow 10} \log x\) (b) \(\lim _{x \rightarrow-1} \log x\) (c) \(\lim _{x \rightarrow 0} \log x\)

Step-by-Step Solution

Answer

(a) \( \log 10 \); (b) does not exist; (c) undefined, but approaches \(-\infty\) as approaching from the right.

Step 1: Understanding the Characteristics of Logarithm Function

The logarithm function \( \log x \) is defined only for positive values of \( x \). This means limits involving negative or zero values inside the log function often result in undefined limits or requiring special consideration.

Step 1: Evaluate the Limit as \( x \rightarrow 10 \)

Since \( x = 10 \) is within the domain of the \( \log x \), and logarithmic functions are continuous for positive \( x \), we can directly substitute the value into the function. Thus, \( \lim _{x \rightarrow 10} \log x = \log 10 \).

Step 2: Evaluate the Limit as \( x \rightarrow -1 \)

Here, \(-1\) is outside the domain of the logarithm function, as the logarithm is undefined for negative \( x \). Therefore, \( \lim _{x \rightarrow -1} \log x \) does not exist.

Step 3: Evaluate the Limit as \( x \rightarrow 0 \)

Similarly, zero is outside the domain of the logarithm function. As \( x \) approaches zero from the right (\( x \rightarrow 0^+ \)), \( \log x \) approaches negative infinity. However, simply stating \( \lim _{x \rightarrow 0} \log x \) without direction gives an undefined limit. In this context, you'd typically write \( \lim _{x \rightarrow 0^+} \log x = -\infty \) if considering approach from the right.

Key Concepts

Logarithm Function

The logarithm function, denoted as \( \log x \), is a mathematical concept used to determine the power to which a specified base must be raised to obtain a given number. In simpler terms, if \( b^y = x \), then \( \log_b x = y \). By default, "log" refers to a logarithm with base 10. The function is only defined for positive real numbers, which has significant implications for limits involving \( \log x \).

Consider the limit \( \lim _{x \rightarrow 10} \log x \). Since 10 is positive, and the logarithm function is continuous for positive inputs, the limit can easily be evaluated by direct substitution. Therefore, \( \lim _{x \rightarrow 10} \log x = \log 10 \). This is straightforward because both the function \( \log x \) and its limit are within the domain of definition.

Domain and Range

In mathematics, the domain of a function refers to all the possible input values for which the function is defined. For the logarithm function \( \log x \), the domain is strictly the set of all positive real numbers. This means any negative or zero values for \( x \) are not within this domain and thus result in undefined expressions.

The range of \( \log x \) is the set of all real numbers, as the output can extend from negative infinity to positive infinity, depending on the values of \( x \).

For example, when evaluating \( \lim_{x \rightarrow -1} \log x \), \( -1 \) is not in the domain of the logarithm function, which is why this limit does not exist. Similarly, evaluating \( \lim _{x \rightarrow 0} \log x \) is undefined in its entirety, but when moving towards zero from the positive side, the function approaches negative infinity.

Undefined Limits

Undefined limits occur when the value(s) that \( x \) approaches fall outside the domain of the function in question. In the realm of logarithms, these scenarios arise when inputs include zero or negative values, which are not part of the function's domain. These points cause the limit to not exist or to require careful consideration.

For instance, in the case of \( \lim _{x \rightarrow -1} \log x \), since \(-1\) is not within the domain, this limit remains undefined. However, undefined does not always mean we can't describe the behavior of the function:

• If evaluating \( \lim_{x \rightarrow 0^{-}} \log x \), it is undefined because logarithms can't directly handle negative inputs.
• If approaching zero from the positive side, \( \lim_{x \rightarrow 0^{+}} \log x \) we find that the function trends towards \(-\infty\).

This behavior highlights the importance of understanding the direction of approach and domain limitations in calculus while dealing with logarithmic functions.