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)]\neq 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 $x = a$.