Function whose jump (non-negative difference of `LHL and RHL` ) of discontinuity is greater than or equal to one. is/are

Options

A. `f(x) = {{:(((e^(1//x)+1))/(e^(1//x)-1)",",x lt 0),(((1 - cos x))/(x)",",x gt 0):}`
B. `g(x) = {{:(((x^(1//3)-1))/(x^(1//2)-1)",",x gt 0),(("In x")/((x -1))",",(1)/(2) lt x lt 1):}`
C. `u(x) = {{:((sin^(-1)2x)/(tan^(-1)3x)",",x in (0, (1)/(2)]),((|sin x|)/(x)",",x lt 0):}`
D. `v(x) = {{:(log_(3)(x+2)",",x gt 2),(log_(1//2)(x^(2)+5)",",x lt 2):}`

Answer

To solve the problem, we need to analyze the given functions and determine if the jump (the non-negative difference between the left-hand limit (LHL) and the right-hand limit (RHL)) of discontinuity is greater than or equal to 1.

### Step-by-Step Solution:

1. **Identify the Functions:**
We have three functions to analyze:
- \( f(x) = e^{1/x} + \frac{1}{e^{1/x}} - 1 \) for \( x < 0 \)
- \( g(x) = \frac{x^{1/3} - 1}{x^{1/2} - 1} \) for \( x = 0 \)
- \( u(x) = \frac{\sin^{-1}(2x)}{\tan^{-1}(3x)} \) for \( x = 0 \)
- \( v(x) = \log_{1/2}(x^2 + 5) \) for \( x < 2 \)

2. **Calculate LHL and RHL for Each Function:**

**For \( f(x) \):**
- **LHL:**
\[ \text{LHL} = \lim_{h \to 0^-} \left( e^{1/h} + \frac{1}{e^{1/h}} - 1 \right) = 0 + \infty - 1 = \infty \]
- **RHL:**
\[ \text{RHL} = \lim_{h \to 0^+} \left( \frac{1 - \cos h}{h} \right) = 0 \]
- **Jump:**
\[ \text{Jump} = \text{LHL} - \text{RHL} = \infty - 0 = \infty \quad (\text{greater than } 1) \]

**For \( g(x) \):**
- **LHL:**
\[ \text{LHL} = \lim_{h \to 0} \frac{h^{1/3} - 1}{h^{1/2} - 1} = 1 \]
- **RHL:**
\[ \text{RHL} = \text{not applicable since } g(x) \text{ is not defined for } x > 0. \]
- **Jump:**
\[ \text{Jump} = \text{RHL} - \text{LHL} \text{ (not applicable)} \]

**For \( u(x) \):**
- **LHL:**
\[ \text{LHL} = \lim_{h \to 0^-} \frac{\sin^{-1}(2h)}{\tan^{-1}(3h)} = -1 \]
- **RHL:**
\[ \text{RHL} = \lim_{h \to 0^+} \frac{\sin^{-1}(2h)}{\tan^{-1}(3h)} = 1 \]
- **Jump:**
\[ \text{Jump} = \text{RHL} - \text{LHL} = 1 - (-1) = 2 \quad (\text{greater than } 1) \]

**For \( v(x) \):**
- **LHL:**
\[ \text{LHL} = \lim_{h \to 2^-} \log_{1/2}(h^2 + 5) = \log_{1/2}(9) = -2 \log(3) \]
- **RHL:**
\[ \text{RHL} = \lim_{h \to 2^+} \log_{3}(h + 2) = \log_{3}(4) \]
- **Jump:**
\[ \text{Jump} = \text{RHL} - \text{LHL} = \log_{3}(4) + 2 \log(3) \quad (\text{check if this is } \geq 1) \]

3. **Conclusion:**
The functions \( f(x) \) and \( u(x) \) have jumps greater than or equal to 1. The function \( g(x) \) is not applicable, and \( v(x) \) needs further evaluation.