Question

If $ \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 $x\to a$ and it is equal to $Im$.
• D. has a limit when $x\to a$ a but it is not equal to $Im$.

Answer 1

Answer: (C)

$\lim_{x\to a-}\left[f\left(x\right) -g\left(x\right)\right] = \lim _{x\to a-} f\left(x\right) = \lim _{x\to a-}g\left(x\right) = ml$
$ \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) =lm$
$\therefore \, \lim _{x\to a} f\left(x\right) g\left(x\right) =lm$