Question

In Exercises \(1-10,\) find the points of continuity and the points of discontinuity of the function. Identify each type of discontinuity. $$y=e^{1 / x}$$

Short answer

The function \(y = e^{1/x}\) is continuous for all values of \(x \neq 0\) and is discontinuous at \(x=0\) due to an undefined operation.

Step-by-step solution

Understand the Function

The function \(y = e^{1/x}\) is an exponential function where \(e\) (approximately 2.71828) is raised to the power \(1/x\). Exponential functions are defined and continuous for all real numbers unless explicitly limited by their domain. Therefore, we need to investigate if there's any value of \(x\) that could cause discontinuity in this function.

Identify Points of Continuity

Since exponential functions are defined for all real numbers, this function will be continuous for all \(x\) except possibly when \(x\) is equal to zero because division by zero is undefined. So, the function \(y = e^{1/x}\) is continuous for all \(x \neq 0\) as there are no jumps, breaks, or holes in the function for these values of \(x\).

Identify Points of Discontinuity

When \(x=0\) in the function \(y=e^{1/x}\), it causes a division by zero scenario, which is undefined. Therefore, \(x=0\) is a point of discontinuity or a hole in the function. This is an example of an indeterminate form of type '0 raised to 0' disrupts the function's continuity at the point \(x=0\).