Q. 6 - Calculus

Question

Find punctured intervals on which the function $f\left(x\right)=\frac{1}{x\ln \left(x+2\right)}$is defined, centered around

$\left(\mathrm{a}\right) x=0 \left(\mathrm{b}\right) x=-1 \left(\mathrm{c}\right) x=-1.5$

Step-by-Step Solution

Answer

Part (a). The punctured intervals is $\left(-0.5,1\right)\cup \left(0,0.5\right)$.

Part (b). The punctured intervals is $\left(-1.5,-1\right)\cup \left(-1,-0.5\right)$.

Part (c). The punctured intervals is $\left(-2,-1.5\right)\cup \left(-1.5,-1\right)$.

Step 1. Given information.

The given function is $f\left(x\right)=\frac{1}{x\ln \left(x+2\right)}$.

Part (a) Step 1. Calculation.

For $x=0$

The punctured interval $\left(c-\delta ,c\right)\cup \left(c,c+\delta \right)$ is given by:
$\left(0-0.5,0\right)\cup \left(0,0+0.5\right)=\left(-0.5,1\right)\cup \left(0,0.5\right)$

Part (b) Step 1. Calculation.

For $x=-1$

The punctured interval $\left(c-\delta ,c\right)\cup \left(c,c+\delta \right)$ is given by:

$\left(-1-0.5,-1\right)\cup \left(-1,-1+0.5\right)=\left(-1.5,-1\right)\cup \left(-1,-0.5\right)$

Part (c) Step 1. Calculation.

For $x=-1.5$

The punctured interval $\left(c-\delta ,c\right)\cup \left(c,c+\delta \right)$ is given by:
$\left(-1.5-0.5,-1.5\right)\cup \left(-1.5,-1.5+0.5\right)=\left(-2,-1.5\right)\cup \left(-1.5,-1\right)$