Chapter 1.3, Problem 9E

To determine

The points of discontinuity of the function. Identify the type of

Discontinuity.

Answer to Problem 9E

The function $y=e^{\frac{1}{x}}$ is continuous and has infinite discontinuity.

Explanation of Solution

Given:

  $y=e^{\frac{1}{x}}$

Calculation:

The given function is composition of two functions $f\left(x\right)=e^x$ and $g\left(x\right)=\frac{1}{x}$ .

And both the functions are continuous.

Find the discontinuity point of the function $y=e^{\frac{1}{x}}$:

Set $\frac{1}{x}$ equal to $0$ , to find where the expression is undefined:

  $x=0$

Its only point of discontinuity occurs at $x=0$ where it is undefined.

  $\underset{x\to 0^{+}}{\lim }{e}^0=\infty$ , so, it has infinite discontinuity.

Thus, the function is continuous and has infinite discontinuity.