Chapter 1.3, Problem 10E

To determine

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

Discontinuity.

Answer to Problem 10E

The function $y=\ln \left(x+1\right)$ is continuous and have discontinuity at $x\le -1$ .

Explanation of Solution

Given:

  $y=\ln \left(x+1\right)$

Calculation:

The given function $y=\ln \left(x+1\right)$ is composition of two functions $f\left(x\right)=\ln x$ and $g\left(x\right)=x+1$ . And both functions are continuous.

Find the discontinuity point of the function$y=\ln \left(x+1\right)$:

Set the argument in $\ln \left(x+1\right)$ less than or equal to $0$ to find where the expression is undefined.

  $x+1\le 0$

Subtract $1$ from both sides of the inequality.

  $x\le -1$

The equation is undefined where the denominator equals $0$ , the argument of a square

root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ : $x\le -1$

  $(-\infty ,-1]$

Thus, the function is continuous and have discontinuity at $x\le -1$