Let `f(x)={{:((e^(x)-1)/(x)",",xgt0),(0",",x=0):}` be a real valued function.
Which of the following statements is/are correct?
1. f(x) is right continuous at x=0.
2. f(x) is discontinuous at x=1.
Seletct the correct answer using the code given below.

Options

A. 1 only
B. 2 only
C. Both 1 and 2
D. Neither 1 nor 2

Explanation

To solve the problem, we need to analyze the function \( f(x) \) defined as follows:

\[ f(x) = \begin{cases} \frac{e^x - 1}{x} & \text{if } x > 0 \\ 0 & \text{if } x = 0 \end{cases} \]

We need to check the following statements:

1. \( f(x) \) is right continuous at \( x = 0 \).
2. \( f(x) \) is discontinuous at \( x = 1 \).

### Step 1: Check Right Continuity at \( x = 0 \)

To check if \( f(x) \) is right continuous at \( x = 0 \), we need to evaluate the right-hand limit as \( x \) approaches \( 0 \):

\[ \lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} \frac{e^x - 1}{x} \]

Using L'Hôpital's Rule (since both the numerator and denominator approach \( 0 \) as \( x \to 0 \)):

\[ \lim_{x \to 0^+} \frac{e^x - 1}{x} = \lim_{x \to 0^+} \frac{e^x}{1} = e^0 = 1 \]

Now, we compare this limit with \( f(0) \):

\[ f(0) = 0 \]

Since:

\[ \lim_{x \to 0^+} f(x) = 1 \quad \text{and} \quad f(0) = 0 \]

Thus, \( f(x) \) is **not right continuous** at \( x = 0 \).

### Step 2: Check Discontinuity at \( x = 1 \)

Next, we need to check if \( f(x) \) is discontinuous at \( x = 1 \). We will find the left-hand limit and right-hand limit at \( x = 1 \).

**Right-hand limit as \( x \to 1^+ \)**:

\[ \lim_{x \to 1^+} f(x) = \lim_{x \to 1^+} \frac{e^x - 1}{x} = \frac{e^1 - 1}{1} = e - 1 \]

**Left-hand limit as \( x \to 1^- \)**:

\[ \lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} \frac{e^x - 1}{x} = \frac{e^1 - 1}{1} = e - 1 \]

Now, we also need to find \( f(1) \):

\[ f(1) = \frac{e^1 - 1}{1} = e - 1 \]

Since both the left-hand limit and right-hand limit at \( x = 1 \) are equal to \( e - 1 \) and \( f(1) \) is also equal to \( e - 1 \), we conclude that \( f(x) \) is **continuous** at \( x = 1 \).

### Conclusion

1. The first statement is **false**: \( f(x) \) is not right continuous at \( x = 0 \).
2. The second statement is **false**: \( f(x) \) is continuous at \( x = 1 \).

Thus, the correct answer is that neither statement is correct.