Question

${\lim }_{x\to a+}f(x)=l={\lim }_{x\to a-}g(x)$ and ${\lim }_{x\to a-}f(x)=m={\lim }_{x\to a+}g(x)$ then the function $f(x)-g(x)$

• A. is continuous at $x=a$
• B. is not continuous at $x=a$
• C. has a limit when $x\to a$, but ${\lim }_{x\to a}[f(x)-g(x)]\ne l-m$
• D. has a limit equal to $I-m$ when $x\to a$

Answer 1

Answer: (B)

${\lim }_{x\to a-}\left[f\left(x\right)-g\left(x\right)\right]={\lim }_{x\to a-}f\left(x\right)={\lim }_{x\to a-}g\left(x\right)=m-l$
${\lim }_{x\to a+}\left[f\left(x\right)-g\left(x\right)\right]={\lim }_{x\to a+}f\left(x\right)-{\lim }_{x\to a+}g\left(x\right)=l-m$
Since ${\lim }_{x\to a-}\left(f\left(x\right)-g\left(x\right)\right)\ne {\lim }_{x\to a+}\left(f\left(x\right)-g\left(x\right)\right)$
$\therefore {\lim }_{x\to a}\left(f\left(x\right)-g\left(x\right)\right)$ does not exist
$\therefore f\left(x\right)-g\left(x\right)$ is not continuous at $x=a$.