Question

Find the limit (if it exists). If it does not exist, explain why. $$ \lim _{x \rightarrow 6^{-}} \ln (6-x) $$

Short answer

The limit of the function \(\ln(6-x)\) as \(x\) approaches 6 from the left side is negative infinity

Step-by-step solution

Identify the Function

The function given is \(\ln(6-x)\) and we are asked to find the limit as \(x\) approaches 6 from the left side, symbolically represented as \(x \rightarrow 6^{-}\).

Evaluate the Function

Evaluate the function as \(x\) approaches 6 from the left side . In this case, \(x\) values will be slightly less than 6, such as 5.999. By plugging these values into the function, the result will be the natural logarithm of a very small positive number, which tends towards negative infinity.

Final Step

Since the value of the function approaches negative infinity as \(x\) approaches 6 from the left side, the limit is negative infinity. This means that the function decreases without bound as \(x\) increases towards 6.

Key concepts

Natural Logarithm

When it comes to understanding limits involving logarithms, it's essential to start with the natural logarithm, often denoted as \( \ln(x) \). The natural logarithm is the inverse of the natural exponential function \( e^x \), where \( e \) is Euler's number, approximately 2.71828. This special logarithm has a base of \( e \), and it describes the time needed to reach a certain level of continuous growth.

One of the key properties of the natural logarithm is how it behaves near the value of zero. As the input of \( \ln(x) \) approaches zero from the positive side, the output extends to negative infinity. This behavior is intuitive because one can only take the logarithm of positive numbers; as these numbers get smaller and smaller, the natural logarithm decreases without bound. In mathematical problems involving limits, recognizing this trend is critical to determining how the function behaves as it approaches specific points.

One-Sided Limits

In calculus, the concept of one-sided limits is a fundamental building block. A one-sided limit is the value that a function approaches as the input approaches a particular point from one specific side – either from the left (denoted as \( x \rightarrow c^- \) for a point \( c \) or from the right (denoted as \( x \rightarrow c^+ \) for a point \( c \). It is crucial in cases where the function behaves differently on either side of a point.

One-sided limits are particularly useful when evaluating functions at points where there is a discontinuity, or the function is not defined. In the given exercise, we are dealing with what's called a 'left-hand limit' as \( x \) approaches 6, we are only interested in the values of \( x \) that are just smaller than 6. Understanding the direction of the approach is key to correctly calculating this type of limit.

Negative Infinity

The concept of negative infinity, denoted by \( -\infty \), is an abstraction used in mathematics to describe an unbounded limit to the downside. It is not a real number, but rather a way to express that the values of a function decrease without bound. When a function's limit is negative infinity, it means that the function's value will continue to decrease, without ever halting, as the variable approaches the limit point.

In the context of logarithms, this occurs when we take the natural logarithm of values that approach zero from the positive side. Importantly, negative infinity is not something to reach; it's more of a trend or direction that the function is taking. Recognizing when a function is trending towards negative infinity is essential for correctly interpreting one-sided limits and understanding the function's overall behavior near certain points.