Explain why the function is discontinuous at the given number a.
f(x)=ln l x-2 l a=2
I don't understand how I would explain this.. in the back of the book it said f(2) is not defined. What do they mean by that?
Your f(x) has an "=" sign in it, before the 2. That must be a misprint. You do not define a function using 2 equals signs.
The log of zero, to any base, is not defined. It is -infinity.
If e^a = 0, then a would be ln 0. No matter how many times you multiply e by itself, you cannot get zero. If a is a large negative number, e^a approaches zero but never gets there. Thefore ln 0 doies not exist (or is "undefined")
To explain why the function is discontinuous at a=2, let's first analyze the function f(x) = ln |x-2|.
In this function, the natural logarithm is applied to the absolute value of the quantity (x-2). This means that the expression inside the logarithm should always be positive in order for the function to be defined.
Now, let's consider the value of a=2. When x equals 2, the expression (x-2) becomes 0. Since the natural logarithm is not defined for input values less than or equal to zero, f(2) is not defined. In other words, we cannot evaluate the function at x=2 because the expression inside the logarithm becomes an undefined value.
This is why the function is discontinuous at a=2. Discontinuity occurs when there is a break or interruption in the function's graph due to the function being undefined or having a jump in its values at a particular point. In this case, the function is undefined at x=2, causing a discontinuity.
To explain why the given function is discontinuous at a=2, which means the function is not defined at x=2, we need to analyze the function and its behavior.
The function in question is f(x) = ln |x - 2|. The natural logarithm function, ln(x), is defined only for positive real numbers. However, in this case, the expression |x - 2| represents the absolute value of (x - 2), which means it can be either positive or negative.
When x is less than 2, the expression inside the absolute value becomes negative, such as (-1, -2, -3, etc.), and the natural logarithm function is not defined for negative numbers. Similarly, when x is greater than 2, the expression inside the absolute value becomes positive, such as (3, 4, 5, etc.), and the natural logarithm function is defined for positive numbers.
Now, let's focus on the discontinuity at a=2. When x is exactly equal to 2, the expression inside the absolute value becomes zero: |2 - 2| = 0. In this case, the natural logarithm function is not defined, as ln(0) is undefined.
Therefore, at x=2, the function f(x) = ln |x - 2| is not defined, indicating a discontinuity. This means that the value of the function cannot be determined at x=2, and thus f(2) is not defined.
To summarize, the function is discontinuous at a=2 because the expression inside the absolute value becomes zero, leading to an undefined value for the natural logarithm function.