The question in the image asks about the type of discontinuity at $x = 0$ for several functions. Let’s go through each one:

a) $y = \frac{8}{1 - 2 \cot x}$

• The cotangent function, $\cot x$, is undefined at $x = 0$, so there is a discontinuity here due to the undefined nature of the cotangent function at zero.
• This is a discontinuity of the infinite type (vertical asymptote) because the function approaches infinity as $x$ approaches 0.

b) $y = \frac{1}{x} - \arcsin(x)$

• The function $\frac{1}{x}$ has a discontinuity at $x = 0$, as division by zero is undefined.
• This discontinuity is also of the infinite type due to the behavior of $\frac{1}{x}$ near 0.

c) $y = \frac{\sin\left(\frac{1}{x}\right)}{e^{\frac{1}{x}} + 1}$

• The term $\sin\left(\frac{1}{x}\right)$ oscillates rapidly as $x$ approaches 0, and the expression $e^{\frac{1}{x}}$ grows exponentially large as $x \to 0$. While both terms cause issues near 0, the denominator ensures the function remains bounded.
• The discontinuity is of the oscillatory type, meaning the function exhibits erratic behavior around $x = 0$ without tending to infinity.

d) $y = \frac{e^{ax} - e^{bx}}{x}, \ (a \neq b)$

• This is an indeterminate form of the type $\frac{0}{0}$ at $x = 0$, but applying L'Hôpital's Rule can resolve it:
  • $\lim_{x \to 0} \frac{e^{ax} - e^{bx}}{x} = a - b$, so the function is removable at $x = 0$, meaning there is no discontinuity after redefining the function at $x = 0$.
• Therefore, this is a removable discontinuity.

Summary:

• a) Infinite discontinuity (vertical asymptote).
• b) Infinite discontinuity (vertical asymptote).
• c) Oscillatory discontinuity.
• d) Removable discontinuity.

Would you like more details on any part of this explanation?

Here are 5 related questions for deeper understanding:

1. What is the formal definition of an infinite discontinuity?
2. How can L'Hôpital’s Rule be applied to indeterminate forms?
3. What types of discontinuities exist in real functions?
4. How does the behavior of the cotangent function affect its discontinuities?
5. What is the significance of oscillatory behavior in discontinuities?

Tip: When analyzing discontinuities, always check if the function can be redefined at the point to resolve any indeterminate form or removable discontinuity.