in order f(x)^1/x is continous at x=0 must be defined as

1. f(0)=0
2. f(0)=1/e
3. f(0)=e
4. f(0)=1 answer with solution

To determine the continuity of the function \( f(x) = \frac{1}{x} \) at \( x = 0 \), we first need to understand what it means for a function to be continuous at a point. A function is continuous at a point \( c \) if the following three conditions are met:

• The function \( f(c) \) is defined.
• The limit of \( f(x) \) as \( x \) approaches \( c \) exists.
• The limit of \( f(x) \) as \( x \) approaches \( c \) is equal to \( f(c) \).

In this case, we are interested in the point \( c = 0 \). Let's analyze the function \( f(x) = \frac{1}{x} \) around this point.

Step 1: Function Definition at \( x = 0 \)

For \( f(x) \) to be continuous at \( x = 0 \), we need to define \( f(0) \). The options provided are:

• f(0) = 0
• f(0) = 1/e
• f(0) = e
• f(0) = 1

Step 2: Evaluating the Limit

Next, we need to find the limit of \( f(x) \) as \( x \) approaches 0:

As \( x \) approaches 0 from the positive side (\( x \to 0^+ \)), \( f(x) = \frac{1}{x} \) approaches \( +\infty \). Conversely, as \( x \) approaches 0 from the negative side (\( x \to 0^- \)), \( f(x) \) approaches \( -\infty \). Therefore, we can conclude:

Limit does not exist: Since the left-hand limit and right-hand limit do not equal each other, the overall limit as \( x \) approaches 0 does not exist.

Step 3: Conclusion on Continuity

Since the limit of \( f(x) \) as \( x \) approaches 0 does not exist, we cannot satisfy the second condition for continuity. Therefore, regardless of how we define \( f(0) \) (whether it is 0, \( 1/e \), \( e \), or 1), the function \( f(x) = \frac{1}{x} \) cannot be continuous at \( x = 0 \).

Final Thoughts

In summary, for the function \( f(x) = \frac{1}{x} \) to be continuous at \( x = 0 \), it must be defined at that point, and the limit must exist and equal the function's value at that point. Since the limit does not exist, \( f(x) \) cannot be continuous at \( x = 0 \) regardless of the value assigned to \( f(0) \).