Question

Determine the point of discontinuity of the function $f\left(x\right)=x+\frac{x+2}{|x+2|},$ Also find the jump of the function at this point.

• A. $x=-2$
• B. $Jump=2$
• C. $Jump=1$
• D. $x=-4$

Answer 1

Answer: (A), (B)

$x=-2$
$Jump=2$

$f\left(x\right)=x+\frac{x+2}{|x+2|}$

at $x=-2$

${lim}_{x\to 2^{-}}f\left(x\right)={lim}_{x\to 2^{-}}(x+\frac{x+2}{(x+2)})=-2-1=-3$

${lim}_{x\to 2^{+}}f\left(x\right)={lim}_{x\to 2^{+}}(x+\frac{x+2}{(x+2)})=-2+1=-1$

LHL$\ne$RHL $\therefore$Discontinuous

$f\left(x\right)$ value jumps from $-3$ to $-1$ at $x=-2$

Jump$=-1-(-3)=2$